Invariant Weakly Positive Semidefinite Kernels with Values in Topologically Ordered $*$-Spaces

TL;DR

This paper studies invariant weakly positive semidefinite kernels valued in ordered $*$-spaces, exploring their decompositions and representations, unifying various dilation results for operator-valued kernels and maps on $*$-semigroups.

Contribution

It introduces conditions for $*$-representations of invariant kernels on VE and VH spaces, unifying multiple existing dilation and positivity results.

Findings

Established conditions for $*$-representations on VE and VH spaces.

Unified various dilation theorems for invariant positive semidefinite kernels.

Connected recent results on positive semidefinite maps with operator values.

Abstract

We consider weakly positive semidefinite kernels valued in ordered $*$-spaces with or without certain topological properties, and investigate their linearisations (Kolmogorov decompositions) as well as their reproducing kernel spaces. The spaces of realisations are of VE (Vector Euclidean) or VH (Vector Hilbert) type, more precisely, vector spaces that possess gramians (vector valued inner products). The main results refer to the case when the kernels are invariant under certain actions of $*$-semigroups and show under which conditions $*$-representations on VE-spaces, or VH-spaces in the topological case, can be obtained. Finally we show that these results unify most of dilation type results for invariant positive semidefinite kernels with operator values as well as recent results on positive semidefinite maps on $*$-semigroups with values operators from a locally bounded topological vector space to its conjugate $Z$-dual space, for $Z$ an ordered $*$-space.

Equations

$$\sum _{j,k=1}^n{\overline{\alpha }}_i{\alpha }_j\mathbf{k}(x_i,x_j)\ge 0,n\in \mathbb{N},{\alpha }_1,\dots ,{\alpha }_n\in \mathbb{C},x_1,\dots ,x_n\in X,$$

$$4[x,y]=\sum _{k=0}^3{\mathrm{i}}^k[(x+{\mathrm{i}}^{k}y,x+{\mathrm{i}}^{k}y],x,y\in \mathcal{E},$$

$$[Te,Te{]}_{\mathcal{F}}\le C^2[e,e{]}_{\mathcal{E}},e\in \mathcal{E}.$$

$$\Vert T\Vert =\inf \{C>0\mid [Te,Te{]}_{\mathcal{F}}\le C^2[e,e{]}_{\mathcal{E}},\text{mbox}foralle\in \mathcal{E}\}.$$

$$[Te,f{]}_{\mathcal{F}}=[e,T^{\ast }f{]}_{\mathcal{E}},e\in \mathcal{E},f\in \mathcal{F}.$$

$$[Ae,f]=[e,Af],e,f\in \mathcal{E}.$$

$$[Ae,e]=[e,Ae],e\in \mathcal{E}.$$

$$[Ae,e]\ge 0,e\in \mathcal{E}.$$

$$\Vert A^{\ast }A\Vert =\Vert A{\Vert }^2,A\in {\mathcal{B}}^{\ast }(\mathcal{E}),$$

$$U_j=\{x\in \mathcal{E}\mid [x,x]\in N_j\},j\in \mathcal{J},$$

$$\widetilde{p_j}(h)=p_j([h,h]{)}^{1/2},h\in \mathcal{E},j\in \mathcal{J}.$$

$$p([e,f])\le 4p([e,e]{)}^{1/2}p([f,f]{)}^{1/2}=4\tilde{p}(e)\tilde{p}(f),e,f\in \mathcal{E}.$$

$$[h\pm k,h\pm k]=[h,h]+[k,k]\pm [h,k]\pm [k,h]\ge 0,$$

$$[h,k]+[k,h]\le [h,h]+[k,k],$$

$$0\le [h+k,h+k]\le [h-k,h-k]+[h+k,h+k]=2([h,h]+[k,k]).$$

$$p([h+k,h+k])\leq 2\bigl{(}p([h,h])+p([k,k])\bigr{)}.$$

$$p([e,f])$$

$$\displaystyle\leq\frac{2}{4}\sum_{k=0}^{3}\bigl{(}p([e,e])+p([\mathrm{i}^{k}f,\mathrm{i}^{k}f])\bigr{)}=2\bigl{(}p([e,e])+p([f,f])\bigr{)}.$$

$$p([e,f])\leq 2\bigl{(}\lambda p([e,e])+\lambda^{-1}p([f,f])\bigr{)},$$

$$p([e,f])\leq\inf_{\lambda>0}2\bigl{(}\lambda p([e,e])+\lambda^{-1}p([f,f])\bigr{)}=4p([e,e])^{1/2}p([f,f])^{1/2}.\qed$$

$$[f,g{]}_{\mathbf{k}}=\sum _{x,y\in X}\overline{f(x)}g(y)\mathbf{k}(x,y),f,g\in {\mathbb{C}}_0^X.$$

$$[f,g{]}_{\mathbf{k}}=[g,f{]}_{\mathbf{k}}^{\ast },f,g\in {\mathbb{C}}_0^X.$$

$$(Kg)(x)=\sum _{y\in X}g(y)\mathbf{k}(x,y),f\in {\mathbb{C}}_0^X.$$

$$[f,g{]}_{\mathbf{k}}=\sum _{x\in X}\overline{f(x)}(Kg)(x),f,g\in {\mathbb{C}}_0^X.$$

$$\sum _{j,k=1}^n\overline{t_k}{t}_j\mathbf{k}(x_k,x_j)\ge 0.$$

$$|s{|}^2\mathbf{k}(x,x)+|t{|}^2\mathbf{k}(y,y)+\overline{s}t\mathbf{k}(x,y)+s\overline{t}\mathbf{k}(y,x)\ge 0.$$

$$\overline{s}t\mathbf{k}(x,y)+s\overline{t}\mathbf{k}(y,x)=\overline{t}s\mathbf{k}(x,y{)}^{\ast }+\overline{s}t\mathbf{k}(y,x{)}^{\ast }.$$

$$\overline{s}t\mathbf{k}(x,y)+s\overline{t}\mathbf{k}(y,x)\ge -|t{|}^2\mathbf{k}(y,y).$$

$$\overline{s}t\mathbf{k}(x,y)+s\overline{t}\mathbf{k}(y,x)=0.$$

$$[\sum _{j=1}^m{t}_{j}V(x_j),\sum _{k=1}^m{s}_{k}V(y_k){]}_{\mathcal{E}}$$

$$=\sum _{j=1}^m\sum _{k=1}^n{s}_k\overline{t_j}[V^{\prime }(x_j),V^{\prime }(y_k){]}_{{\mathcal{E}}^{\prime }}=[\sum _{j=1}^m{t}_j{V}^{\prime }(x_j),\sum _{k=1}^n{s}_k{V}^{\prime }(y_k){]}_{{\mathcal{E}}^{\prime }},$$

Full text

The second named author’s work supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0119.

1. Introduction

The dilation theory, initiated by the seminal articles of M.A. Naĭmark in [23] and [24], consists today of an extraordinary large diversity of results that may look, at the first glance, as having next to nothing in common, e.g. see N. Aronszajn [1], W.B. Arveson [2], S.D. Barreto et al. [5], D. Gaşpar and P. Gaşpar [10], [9], A. Gheondea and B.E. Uğurcan [12], J. Górniak and A. Weron [13], [14], J. Heo [15], G.G. Kasparov [17], R.M. Loynes [19], G.J. Murphy [22], M. Skeide [30], W.F. Stinespring [31], F.H. Szafraniec [32], [33], B. Sz.-Nagy [34], to cite a few only. In a series of recent articles [11], [3], and [4], a unification of this theory under some general results of operator valued positive semidefinite kernels that are invariant under actions of $*$-semigroups has been initiated. Historically, based on some classical results on scalar kernels of J. Mercer [21] and A.N. Kolmogorov [18], positive semidefinite kernels that are invariant under actions of groups have been used more than forty years ago in mathematical models of quantum physics by D.E. Evans and J.T. Lewis [8] and in probability theory by K.R. Parthasaraty and K. Schmidt [25] and turned out to be successful even beyond positive semidefiniteness, as in [7].

On the other hand, positive semidefiniteness of a scalar valued kernel $\mathbf{k}\colon X\times X\rightarrow{\mathbb{C}}$, defined as

[TABLE]

has many different ways of generalisation when comes to operator valued kernels and, consequently, the diversity of dilation results increases considerably. From the point of view of unification of dilation theory there is a challenge: are there a concept of positive semidefiniteness and a concept of vector space where these kernels take values, that can yield dilation theorems that are sufficiently general to contain all (most) of the other dilation theorems for operator valued kernels or maps? Clearly, such a concept of ”weakly” positive semidefiniteness must refer simply to the bare situation as in (1.1), while the concept of vector space should be an ordered $*$-spaces and, consequently, the spaces of dilation that we expect should be, in the nontopological case, of VE (Vector Euclidean) type or, in the topological case, of VH (Vector Hilbert) type, in the sense of R.M. Loynes [20]. So far, ”weakly” positive semidefiniteness have been rarely considered, e.g. W.L. Paschke [26] has a remark on maps on $C^{*}$-algebras and, for the special purposes of reproducing kernel spaces, it was first considered in [9] and then used in [27] as well.

The aim of this article is to develop a systematic study of invariant weakly positive semidefinite kernels with values in ordered $*$-spaces and to show that most of the previous dilation results as in [11], [3], [4] and hence, most of the known dilation theory, can be recovered under this setting. The main results are contained in theorems 4.3, 4.5, and 4.6, from which we then show how special cases concerning different kinds of ”stronger” positive semidefiniteness can be derived. Of course, since these dilation theorems are so general, in each particular case we expect that some additional technical difficulties should show up, but here the main idea is of unification and there is always a price to be paid in this enterprise.

In the following we briefly present the contents of this article. In Section 2 we briefly recall the general terminology on ordered $*$-spaces and their topological versions, on VE-spaces (Vector Euclidean spaces) and VH-spaces (Vector Hilbert spaces) and their operator theory. In Lemma 2.2 we prove a surrogate of Schwarz inequality which turns out to be very useful. This inequality with the constant $2$ has been claimed before in [6], or simply stated without proof or reference as in [27], but since the proofs we have seen until now turned out to be flawed, we give a detailed proof of it with constant $4$.

Section 3 contains a detailed study of linearisations and reproducing kernel spaces associated to weakly positive semidefinite kernels which pertains to VE-spaces or VH-spaces. Since the geometry of these spaces is so badly behaved, a careful treatment is necessary from the point of view of minimality and of the equivalence of linearisation with reproducing kernel space. The main results are contained in Section 4. In Theorem 4.3 we obtain the nontopological fabric of dilation theorems for invariant weakly positive semidefinite kernels with values in ordered $*$-spaces and then we obtain two topological versions, Theorem 4.5 for bounded operators and Theorem 4.6 for continuously adjointable operators. As expected, both of these topological variants refer to a variant of the B. Sz.-Nagy boundedness condition but, it is interesting to observe that, a second boundedness condition which refers to an anomaly of operator theory for continuously adjointable operators on VH-spaces related to the continuity of the adjoint, see condition (c) in Theorem 5.3, does not show up.

Finally, in Section 5 we show that the main theorems contain the dilation results obtained in [11], [3], and [4], and hence most of the dilation theory, by explicitly showing how to put the stage in each case. A special observation is that for the reproducing kernel space versions, which is one of the main tool we use, there are some technical difficulties related to missing a version of Riesz’s Representation Theorem in VH-spaces or VE-spaces and which is solved by carefully using identifications. In addition, we show how the recent results of F. Pater and T. Bînzar [27] on positive semidefinite maps on $*$-semigroups with values operators from a vector space to its conjugate $Z$-dual space, for $Z$ an ordered $*$-space, that generalise previous results of J. Górniak and A. Weron [14], can be recovered by our main results, with actually stronger statements.

2. Notation and Preliminary Results

In this section we review some of the definitions and some theorems on ordered $*$-spaces, topologically ordered $*$-spaces, admissible spaces, VE-spaces, topologically VE-spaces and VH-spaces, and their operator theory, cf. R.M. Loynes, [19], [20] and, for a modern treatment of the subject and some proofs, we refer also to [3] and [4].

2.1. Topologically Ordered $*$-Spaces.

A complex vector space $Z$ is called an ordered $*$-space if:

• (a1)

$Z$ has an involution $*$, that is, a map $Z\ni z\mapsto z^{*}\in Z$ that is conjugate linear (($sx+ty)^{*}=\overline{s}x^{*}+\overline{t}y^{*}$ for all $s,t\in{\mathbb{C}}$ and all $x,y\in Z$) and involutive ($(z^{*})^{*}=z$ for all $z\in Z$).

• (a2)

In $Z$ there is a convex cone $Z_{+}$ ($sx+ty\in Z_{+}$ for all numbers $s,t\geq 0$ and all $x,y\in Z_{+}$), that is strict ($Z_{+}\cap-Z_{+}=\{0\}$), and consisting of selfadjoint elements only ($z^{*}=z$ for all $z\in Z_{+}$). This cone is used to define a partial order in $Z$ by: $z_{1}\geq z_{2}$ if $z_{1}-z_{2}\in Z_{+}$.

The complex vector space $Z$ is called a topologically ordered $*$-space if it is an ordered $*$-space and:

• (a3)

$Z$ is a Hausdorff separated locally convex space.

• (a4)

The cone $Z_{+}$ is closed, with respect to this topology.

• (a5)

The topology of $Z$ is compatible with the partial ordering in the sense that there exists a base of the topology, linearly generated by a family of neighbourhoods $\{N_{j}\}_{j\in{\mathcal{J}}}$ of the origin, such that all of them are absolutely convex and solid, that is, whenever $x\in N_{j}$ and $0\leq y\leq x$ then $y\in N_{j}$.

It can be proven that axiom (a5) is equivalent with the following one, see [4]:

• (a5′)

There exists a collection of seminorms $\{p_{j}\}_{j\in{\mathcal{J}}}$ defining the topology of $Z$ that are increasing, that is, $0\leq x\leq y$ implies $p_{j}(x)\leq p_{j}(y)$.

We denote the collection of all increasing continuous seminorms on $Z$ by $S(Z)$. $Z$ is called an admissible space if, in addition to the axioms (a1)–(a5),

• (a6)

The topology on $Z$ is complete.

2.2. Vector Euclidean Spaces and Their Linear Operators.

Given a complex linear space ${\mathcal{E}}$ and an ordered $*$-space $Z$, a $Z$-valued inner product or $Z$-gramian is, by definition, a mapping ${\mathcal{E}}\times{\mathcal{E}}\ni(x,y)\mapsto[x,y]\in Z$ subject to the following properties:

• (ve1)

$[x,x]\geq 0$ for all $x\in{\mathcal{E}}$, and $[x,x]=0$ if and only if $x=0$.

• (ve2)

$[x,y]=[y,x]^{*}$ for all $x,y\in{\mathcal{E}}$.

• (ve3)

$[x,ay_{1}+by_{2}]=a[x,y_{1}]+b[x,y_{2}]$ for all $a,b\in\mathbb{C}$ and all $x_{1},x_{2}\in{\mathcal{E}}$.

A complex linear space ${\mathcal{E}}$ onto which a $Z$-valued inner product $[\cdot,\cdot]$ is specified, for a certain ordered $*$-space $Z$, is called a VE-space (Vector Euclidean space) over $Z$.

In any VE-space ${\mathcal{E}}$ over an ordered $*$-space $Z$ the familiar polarisation formula

[TABLE]

holds, which shows that the $Z$-valued inner product is perfectly defined by the $Z$-valued quadratic map ${\mathcal{E}}\ni x\mapsto[x,x]\in Z$.

The concept of VE-spaces isomorphism is also naturally defined: this is just a linear bijection $U\colon{\mathcal{E}}\rightarrow{\mathcal{F}}$, for two VE-spaces over the same ordered $*$-space $Z$, which is isometric, that is, $[Ux,Uy]_{\mathcal{F}}=[x,y]_{\mathcal{E}}$ for all $x,y\in{\mathcal{E}}$.

A useful result for the constructions in this paper is the following lemma.

Lemma 2.1** (Loynes [19]).**

Let $Z$ be an ordered $*$-space, ${\mathcal{E}}$ a complex vector space and $[\cdot,\cdot]\colon{\mathcal{E}}\times{\mathcal{E}}\rightarrow Z$ a positive semidefinite sesquilinear map, that is, $[\cdot,\cdot]$ is linear in the second variable, conjugate linear in the first variable, and $[x,x]\geq 0$ for all $x\in{\mathcal{E}}$. If $f\in{\mathcal{E}}$ is such that $[f,f]=0$, then $[f,f^{\prime}]=[f^{\prime},f]=0$ for all $f^{\prime}\in{\mathcal{E}}$.

Given two VE-spaces ${\mathcal{E}}$ and ${\mathcal{F}}$, over the same ordered $*$-space $Z$, one can consider the vector space ${\mathcal{L}}({\mathcal{E}},{\mathcal{F}})$ of all linear operators $T\colon{\mathcal{E}}\rightarrow{\mathcal{F}}$. The operator $T$ is called bounded if there exists $C\geq 0$ such that

[TABLE]

Note that the inequality (2.2) is in the sense of the order of $Z$ uniquely determined by the cone $Z_{+}$, see the axiom (a2). The infimum of these scalars is denoted by $\|T\|$ and it is called the operator norm of $T$, more precisely,

[TABLE]

Let ${\mathcal{B}}({\mathcal{E}},{\mathcal{F}})$ denote the collection of all bounded linear operators $T\colon{\mathcal{E}}\rightarrow{\mathcal{F}}$. Then ${\mathcal{B}}({\mathcal{E}},{\mathcal{F}})$ is a linear space and $\|\cdot\|$ is a norm on it, cf. Theorem 1 in [20]. In addition, if $T$ and $S$ are bounded linear operators acting between appropriate VE-spaces over the same ordered $*$-space $Z$, then $\|TS\|\leq\|T\|\|S\|$, in particular $TS$ is bounded. If ${\mathcal{E}}={\mathcal{F}}$ then ${\mathcal{B}}({\mathcal{E}})={\mathcal{B}}({\mathcal{E}},{\mathcal{E}})$ is a normed algebra, more precisely, the operator norm is submultiplicative.

A linear operator $T\in{\mathcal{L}}({\mathcal{E}},{\mathcal{F}})$ is called adjointable if there exists $T^{*}\in{\mathcal{L}}({\mathcal{F}},{\mathcal{E}})$ such that

[TABLE]

The operator $T^{*}$, if it exists, is uniquely determined by $T$ and called its adjoint. Since an analog of the Riesz Representation Theorem for VE-spaces does not exist, in general, there may be not so many adjointable operators. We denote by ${\mathcal{L}}^{*}({\mathcal{E}},{\mathcal{F}})$ the vector space of all adjointable operators from ${\mathcal{L}}({\mathcal{E}},{\mathcal{F}})$. Note that ${\mathcal{L}}^{*}({\mathcal{E}})={\mathcal{L}}^{*}({\mathcal{E}},{\mathcal{E}})$ is a $*$-algebra with respect to the involution $*$ determined by the operation of taking the adjoint.

An operator $A\in{\mathcal{L}}({\mathcal{E}})$ is called selfadjoint if

[TABLE]

Clearly, any selfadjoint operator $A$ is adjointable and $A=A^{*}$. By the polarisation formula (2.1), $A$ is selfadjoint if and only if

[TABLE]

An operator $A\in{\mathcal{L}}({\mathcal{E}})$ is positive if

[TABLE]

Since the cone $Z_{+}$ consists of selfadjoint elements only, any positive operator is selfadjoint and hence adjointable.

Let ${\mathcal{B}}^{*}({\mathcal{E}})$ denote the collection of all adjointable bounded linear operators $T\colon{\mathcal{E}}\rightarrow{\mathcal{E}}$. Then ${\mathcal{B}}^{*}({\mathcal{E}})$ is a pre-$C^{*}$-algebra, that is, it is a normed $*$-algebra with the property

[TABLE]

cf. Theorem 4 in [20]. In particular, the involution $*$ is isometric on ${\mathcal{B}}^{*}({\mathcal{E}})$, that is, $\|A^{*}\|=\|A\|$ for all $A\in{\mathcal{B}}^{*}({\mathcal{E}})$.

If $A\in{\mathcal{B}}^{*}({\mathcal{E}})$ can be factored $A=T^{*}T$, for some $T\in{\mathcal{B}}^{*}({\mathcal{E}})$, then $A$ is positive. If, in addition, ${\mathcal{B}}^{*}({\mathcal{E}})$ is complete, and hence a $C^{*}$-algebra, and $A\in{\mathcal{B}}^{*}({\mathcal{E}})$ is positive, then $A=T^{*}T$ for some $T\in{\mathcal{B}}^{*}({\mathcal{E}})$, cf. Lemma 2 in [20].

2.3. Vector Hilbert Spaces and Their Linear Operators.

If $Z$ is a topologically ordered $*$-space, any VE-space ${\mathcal{E}}$ can be made in a natural way into a Hausdorff separated locally convex space by considering the weakest locally convex topology on ${\mathcal{E}}$ that makes the mapping $E\ni h\mapsto[h,h]\in Z$ continuous, more precisely, letting $\{N_{j}\}_{j\in{\mathcal{J}}}$ be the collection of convex and solid neighbourhoods of the origin in $Z$ as in axiom (a5), the collection of sets

[TABLE]

is a topological base of neighbourhoods of the origin of ${\mathcal{E}}$ that linearly generates the weakest locally convex topology on ${\mathcal{E}}$ that makes all mappings ${\mathcal{E}}\ni h\mapsto[h,h]\in Z$ continuous, cf. Theorem 1 in [19]. In terms of seminorms, this topology can be defined in the following way: let $\{p_{j}\}_{j\in{\mathcal{J}}}$ be a family of increasing seminorms defining the topology of $Z$ as in axiom (a5′) and let

[TABLE]

Then each $\tilde{p_{j}}$ is a seminorm on ${\mathcal{E}}$ and its topology is fully determined by the family $\{\tilde{p_{j}}\}_{j\in{\mathcal{J}}}$, see Lemma 1.3 in [4]. With respect to this topology, we call ${\mathcal{E}}$ a topological VE-space over $Z$.

We first prove a surrogate of Schwarz Inequality that we will use several times in this article.

Lemma 2.2**.**

Let ${\mathcal{E}}$ be a topological VE-space over the topologically ordered $*$-space $Z$ and $p\in S(Z)$. Then

[TABLE]

Proof.

For arbitrary $h,k\in{\mathcal{E}}$ we have

[TABLE]

in particular,

[TABLE]

and

[TABLE]

Taking into account that $p\in S(Z)$ is increasing, from (2.12) it follows that

[TABLE]

Let now $e,f\in{\mathcal{E}}$ be arbitrary. By the polarisation formula (2.1) and (2.13), we have

[TABLE]

Letting $\lambda>0$ be arbitrary and changing $e$ with $\sqrt{\lambda}e$ and $f$ with $f/\sqrt{\lambda}$ in the previous inequality, we get

[TABLE]

hence, since the left hand side does not depend on $\lambda$, it follows

[TABLE]

Let ${\mathcal{E}}$ and ${\mathcal{F}}$ be two topological VE-spaces over the same topologically ordered $*$-space $Z$. Clearly, any bounded linear operator $T\colon{\mathcal{E}}\rightarrow{\mathcal{F}}$, with definition as in (2.3), is continuous.

If both of $Z$ and ${\mathcal{E}}$ are complete with respect to their specified locally convex topologies, then ${\mathcal{E}}$ is called a VH-space (Vector Hilbert space). Any topological VE-space ${\mathcal{E}}$ on a topologically ordered $*$-space can be embedded as a dense subspace of a VH-space ${\mathcal{H}}$, uniquely determined up to an isomorphism, cf. Theorem 2 in [19]. Note that, given two VH-spaces ${\mathcal{H}}$ and ${\mathcal{K}}$, over the same admissible space $Z$, any isomorphism $U\colon{\mathcal{H}}\rightarrow{\mathcal{K}}$ in the sense of VE-spaces, is automatically bounded and adjointable, hence $U\in{\mathcal{B}}({\mathcal{H}},{\mathcal{K}})$, and it is natural to call this operator unitary.

If ${\mathcal{E}}$ and ${\mathcal{F}}$ are topological VE-spaces over the same admissible space $Z$ and ${\mathcal{F}}$ is complete, that is, a VH-space, then ${\mathcal{B}}({\mathcal{E}},{\mathcal{F}})$ is a Banach space, with respect to the operator norm. In particular, if ${\mathcal{E}}$ is a VH-space, then ${\mathcal{B}}^{*}({\mathcal{E}})$ is a $C^{*}$-algebra. Note that, in this case, the usual notion of $C^{*}$-algebra positive elements in ${\mathcal{B}}^{*}({\mathcal{E}})$ coincides with that of positive operators in the sense of (2.7), [19].

For topological VE-spaces ${\mathcal{E}}$ and ${\mathcal{F}}$ over the same topologically ordered $*$-space $Z$, we denote the space of all linear continuous operators $T\colon{\mathcal{E}}\rightarrow{\mathcal{F}}$ by ${\mathcal{L}}_{\mathrm{c}}({\mathcal{E}},{\mathcal{F}})$, and in particular, ${\mathcal{L}}_{\mathrm{c}}({\mathcal{E}},{\mathcal{E}})$ by ${\mathcal{L}}_{\mathrm{c}}({\mathcal{E}})$. The $*$-algebra of all continuous and continuously adjointable linear operators $T\colon{\mathcal{E}}\rightarrow{\mathcal{F}}$ are denoted by ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{E}},{\mathcal{F}})$, and ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{E}})={\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{E}},{\mathcal{E}})$, see [4].

A subspace ${\mathcal{M}}$ of a VH-space ${\mathcal{H}}$ is orthocomplemented or accessible if every element $x\in{\mathcal{H}}$ can be written as $x=y+z$ where $y\in{\mathcal{M}}$ and $z$ is such that $[z,m]=0$ for all $m\in{\mathcal{M}}$, that is, $z$ is in the orthogonal companion ${\mathcal{M}}^{\bot}$ of ${\mathcal{M}}$. If such a decomposition exists it is unique. Also, any orthocomplemented subspace is closed. A closed subspace ${\mathcal{M}}$ of ${\mathcal{H}}$ is orthocomplemented if and only if it is the range of a selfadjoint projection, that is, an adjointable linear operator $P\colon{\mathcal{H}}\rightarrow{\mathcal{H}}$ such that $P^{2}=P=P^{*}$. Note that any selfadjoint projection is a contraction, in particular it is bounded.

3. Weakly Positive Semidefinite Kernels

3.1. Hermitian Kernels.

Let $X$ be a nonempty set and $Z$ an ordered $*$-space. A map $\mathbf{k}\colon X\times X\rightarrow Z$ is called a $Z$-valued kernel on $X$. If no confusion may arise we also say simply that $\mathbf{k}$ is a kernel. The adjoint kernel $\mathbf{k}^{*}\colon X\times X\rightarrow Z$ is defined by $\mathbf{k}^{*}(x,y)=\mathbf{k}(y,x)^{*}$, for $x,y\in X$. The kernel $\mathbf{k}$ is called Hermitian if $\mathbf{k}^{*}=\mathbf{k}$.

Consider ${\mathbb{C}}^{X}$ the complex vector space of all functions $f\colon X\rightarrow{\mathbb{C}}$, as well as its subspace ${\mathbb{C}}^{X}_{0}$ consisting of those functions $f\in{\mathbb{C}}^{X}$ with finite support. Given a $Z$-valued kernel $\mathbf{k}$ on $X$, a pairing $[\cdot,\cdot]_{\mathbf{k}}\colon{\mathbb{C}}_{0}^{X}\times{\mathbb{C}}_{0}^{X}\rightarrow Z$ can be defined

[TABLE]

The pairing $[\cdot,\cdot]_{\mathbf{k}}$ is linear in the second variable and conjugate linear in the first variable. If, in addition, $\mathbf{k}=\mathbf{k}^{*}$, then the pairing $[\cdot,\cdot]_{\mathbf{k}}$ is Hermitian, that is,

[TABLE]

Conversely, if the pairing $[\cdot,\cdot]_{\mathbf{k}}$ is Hermitian then $\mathbf{k}=\mathbf{k}^{*}$.

A convolution operator $K\colon{\mathbb{C}}^{X}_{0}\rightarrow Z^{X}$, where $Z^{X}$ is the complex vector space of all functions $g\colon X\rightarrow Z$, can be associated to the $Z$-kernel $\mathbf{k}$ by

[TABLE]

Clearly, $K$ is a linear operator. A natural relation exists between the paring $[\cdot,\cdot]_{\mathbf{k}}$ and the convolution operator $K$, more precisely,

[TABLE]

Therefore, it is easy to see from here that the kernel $\mathbf{k}$ is Hermitian if and only if the pairing $[\cdot,\cdot]_{\mathbf{k}}$ is Hermitian.

Given a natural number $n$, a $Z$-valued kernel $\mathbf{k}$ is called weakly $n$-positive if for all $x_{1},\ldots,x_{n}\in X$ and all $t_{1},\ldots,t_{n}\in{\mathbb{C}}$ we have

[TABLE]

The kernel $\mathbf{k}$ is called weakly positive semidefinite if it is $n$-positive for all $n\in{\mathbb{N}}$.

Lemma 3.1**.**

Let the $Z$-kernel $\mathbf{k}$ on $X$ be weakly $2$-positive. Then:

(1)* $\mathbf{k}$ is Hermitian.*

(2)* If, for some $x\in X$, $\mathbf{k}(x,x)=0$, then $\mathbf{k}(x,y)=0$ for all $y\in X$.*

(3)* There exists a unique decomposition $X=X_{0}\cup X_{1}$, $X_{0}\cap X_{1}=\emptyset$, such that $\mathbf{k}(x,y)=0$ for all $x,y\in X_{0}$ and $\mathbf{k}(x,x)\neq 0$ for all $x\in X_{1}$.*

Proof.

(1) Clearly, weak $2$-positivity implies weak $1$-positivity, hence $\mathbf{k}(x,x)\geq 0$ for all $x\in X$. Let $x,y\in X$ be arbitrary. Since $\mathbf{k}$ is weakly $2$-positive, for any $s,t\in{\mathbb{C}}$ we have

[TABLE]

Since the sum of the first two terms in (3.6) is in $Z_{+}$ and taking into account that $Z_{+}$ consists of selfadjoint elements only, it follows that the sum of the last two terms in (3.6) is selfadjoint, that is,

[TABLE]

Letting $s=t=1$ and then $s=1$ and $t=\mathrm{i}$, it follows that $\mathbf{k}(y,x)=\mathbf{k}(x,y)^{*}$.

(2) Assume that $\mathbf{k}(x,x)=0$ and let $y\in X$ be arbitrary. From (3.6) it follows that for all $s,t\in{\mathbb{C}}$ we have

[TABLE]

We claim that for all $s,t\in{\mathbb{C}}$ we have

[TABLE]

To prove this, note that for $t=0$ the equality (3.8) it trivially true. If $t\in{\mathbb{C}}\setminus\{0\}$, note that we can distinguish two cases: first, if $\mathbf{k}(y,y)=0$, then from (3.7) it follows $\overline{s}t\mathbf{k}(x,y)+s\overline{t}\mathbf{k}(y,x)\geq 0$ and then, changing $t$ to $-t$ the opposite inequality holds, hence (3.8). The second case is $\mathbf{k}(y,y)\neq 0$ when we observe that the right hand side in (3.7) does not depend on $s$ hence, replacing $s$ by $ns$, $n\in{\mathbb{Z}}$, a routine reasoning shows that (3.8) must hold as well.

Finally, in (3.8) we first let $s=1=t$ and then $s=1$ and $t=\mathrm{i}$ and solve for $\mathbf{k}(x,y)$ which should be [math].

(3) Denote $X_{0}=\{x\in X\mid\mathbf{k}(x,x)=0\}$ and let $X_{1}=X\setminus X_{0}$. Then use (2) in order to obtain $\mathbf{k}(x,y)=0$ for all $x,y\in X_{0}$. ∎

3.2. Weak Linearisations

Given an ordered $*$-space $Z$ and a $Z$-valued kernel $\mathbf{k}$ on a nonempty set $X$, a weak VE-space linearisation, or weak Kolmogorov decomposition of $\mathbf{k}$ is, by definition, a pair $({\mathcal{E}};V)$, subject to the following conditions:

• (vel1)

${\mathcal{E}}$ is a VE-space over the ordered $*$-space $Z$.

• (vel2)

$V\colon X\rightarrow{\mathcal{E}}$ satisfies $\mathbf{k}(x,y)=[V(x),V(y)]_{\mathcal{E}}$ for all $x,y\in X$.

If, in addition, the following condition holds

• (vel3)

$\operatorname{Lin}V(X)={\mathcal{E}}$,

then the weak VE-space linearisation $({\mathcal{E}};V)$ is called minimal.

Two weak VE-space linearisations $(V;{\mathcal{E}})$ and $(V^{\prime};{\mathcal{E}}^{\prime})$ of the same kernel $\mathbf{k}$ are called unitarily equivalent if there exists a unitary operator $U\colon{\mathcal{E}}\rightarrow{\mathcal{E}}^{\prime}$ such that $UV(x)=V^{\prime}(x)$ for all $x\in X$.

Remarks 3.2**.**

(1) Note that any two minimal weak VE-space linearisations $({\mathcal{E}};V)$ and $({\mathcal{E}}^{\prime};V^{\prime})$ of the same $Z$-kernel $\mathbf{k}$ are unitarily equivalent. The proof follows in the usual way: if $({\mathcal{E}}^{\prime};V^{\prime})$ is another minimal weak VE-space linearisation of $\mathbf{k}$, for arbitrary $x_{1},\ldots,x_{m},y_{1},\ldots,y_{n}\in X$ and arbitrary $t_{1},\ldots,t_{m},s_{1},\ldots,s_{n}\in{\mathbb{C}}$, we have

[TABLE]

hence $U\colon\operatorname{Lin}V(X)\rightarrow\operatorname{Lin}V^{\prime}(X)$ defined by

[TABLE]

is a correctly defined linear operator, isometric, everywhere defined, and onto. Thus, $U$ is a VE-space isomorphism $U\colon{\mathcal{E}}\rightarrow{\mathcal{E}}^{\prime}$ and $UV(x)=V^{\prime}(x)$ for all $x\in X$, by construction.

(2) From any weak VE-space linearisation $({\mathcal{E}};V)$ of $\mathbf{k}$ one can make a minimal one in a canonical way, more precisely, letting ${\mathcal{E}}_{0}=\operatorname{Lin}V(X)$ and $V_{0}\colon X\rightarrow{\mathcal{E}}_{0}$ defined by $V_{0}(x)=V(x)$, $x\in X$, it follows that $({\mathcal{E}}_{0};V_{0})$ is a minimal weak VE-space linearisation of $\mathbf{k}$.

Let us assume now that $Z$ is an admissible space and $\mathbf{k}$ is a $Z$-kernel on a set $X$. A weak VH-space linearisation of $\mathbf{k}$ is a linearisation $({\mathcal{H}};V)$ of $\mathbf{k}$ such that ${\mathcal{H}}$ is a VH-space. The weak VH-space linearisation $({\mathcal{H}};V)$ is called topologically minimal if

• (vhl3)

$\operatorname{Lin}V(X)$ is dense in ${\mathcal{H}}$.

Two weak VH-space linearisations $({\mathcal{H}};V)$ and $({\mathcal{H}}^{\prime};V^{\prime})$ of the same $Z$-kernel $\mathbf{k}$ are called unitary equivalent if there exists a unitary operator $U\in{\mathcal{B}}^{*}({\mathcal{H}},{\mathcal{H}}^{\prime})$ such that $UV(x)=V^{\prime}(x)$ for all $x\in X$.

Remarks 3.3**.**

(a) Any two topologically minimal weak VH-space linearisations of the same $Z$-kernel are unitarily equivalent. Indeed, letting $({\mathcal{H}};V)$ and $({\mathcal{H}}^{\prime};V^{\prime})$ be two minimal weak VH-space linearisations of the $Z$-kernel $\mathbf{k}$, we proceed as in Remark 3.2.(a) and define $U\colon\operatorname{Lin}V(X)\rightarrow\operatorname{Lin}V^{\prime}(X)$ as in (3.9). Since $U$ is isometric, it is bounded in the sense of (2.2), hence continuous, and then $U$ can be uniquely extended to an isometric operator $U\colon{\mathcal{H}}\rightarrow{\mathcal{H}}^{\prime}$. Since $\operatorname{Lin}V^{\prime}(X)$ is dense in ${\mathcal{H}}^{\prime}$ and $U$ has closed range, it follows that $U$ is surjective, hence $U\in{\mathcal{B}}^{*}({\mathcal{H}},{\mathcal{H}}^{\prime})$ is unitary and, by its definition, see (3.9), we have $UV(x)=V^{\prime}(x)$ for all $x\in X$.

(b) From any weak VH-space linearisation $({\mathcal{H}};V)$ of $\mathbf{k}$ one can make, in a canonical way, a topologically minimal weak VH-space linearisation $({\mathcal{H}}_{0};V_{0})$ by letting ${\mathcal{H}}_{0}=\overline{\operatorname{Lin}V(X)}$ and $V_{0}(x)=V(x)$ for all $x\in X$.

Theorem 3.4**.**

(a)* Given an ordered $*$-space $Z$ and a $Z$-valued kernel $\mathbf{k}$ on a nonempty set $X$, the following assertions are equivalent:*

• (1)

$\mathbf{k}$* is positive semidefinite.*

• (2)

$\mathbf{k}$* admits a weak VE-space linearisation $({\mathcal{E}};V)$.*

Moreover, if exists, a weak VE-space linearisation $({\mathcal{E}};V)$ can always be chosen such that ${\mathcal{E}}\subseteq Z^{X}$, that is, consisting of functions $f\colon X\rightarrow Z$ only, and minimal.

(b)* If, in addition, $Z$ is an admissible space and $\mathbf{k}\colon X\times X\rightarrow Z$ is a kernel, then any of the assertions (1) and (2) is equivalent with:*

• (3)

$\mathbf{k}$* admits a weak VH-space linearisation $({\mathcal{H}};V)$.*

Moreover, if exists, a weak VH-space linearisation $({\mathcal{H}};V)$ can always be chosen such that ${\mathcal{H}}\subseteq Z^{X}$ and topologically minimal.

Proof.

(1)$\Rightarrow$(2). Assuming that $\mathbf{k}$ is positive semidefinite, by Lemma 3.1.(1) it follows that $\mathbf{k}$ is Hermitian, that is, $\mathbf{k}(x,y)^{*}=\mathbf{k}(y,x)$ for all $x,y\in X$. With notation as in Subsection 3.1, we consider the convolution operator $K\colon{\mathbb{C}}^{X}_{0}\rightarrow Z^{X}$ and let $Z^{X}_{K}$ be its range, more precisely,

[TABLE]

A pairing $[\cdot,\cdot]_{{\mathcal{E}}}\colon Z_{K}^{X}\times Z_{K}^{X}\rightarrow Z$ can be defined by

[TABLE]

where $f=Kh$ and $e=Kg$ for some $g,h\in{\mathbb{C}}^{X}_{0}$. We observe that

[TABLE]

which shows that the definition in (3.11) is correct, that is, independent of $g$ and $h$ such that $e=Kg$ and $f=Kh$.

We claim that $[\cdot,\cdot]_{{\mathcal{E}}}$ is a $Z$-valued inner product, that is, it satisfies all the requirements (ve1)–(ve3). The only fact that needs a proof is $[f,f]_{{\mathcal{E}}}=0$ implies $f=0$. To see this, we use Lemma 2.1 and first get that $[f,f^{\prime}]_{{\mathcal{E}}}=0$ for all $f^{\prime}\in Z^{X}_{K}$. For each $x\in X$, let $\delta_{x}\in{\mathbb{C}}^{X}_{0}$ denote the $\delta$-function with support $\{x\}$,

[TABLE]

Letting $f^{\prime}=K\delta_{x}$ we have

[TABLE]

hence, since $x\in X$ are arbitrary, it follows that $f=0$.

Thus, $(Z^{X}_{K};[\cdot,\cdot]_{{\mathcal{E}}})$ is a VE-space. For each $x\in X$ we define $V(x)\in Z^{X}_{K}\subseteq{\mathcal{E}}$ by

[TABLE]

Actually, there is an even more explicit way of expressing $V(x)$, namely,

[TABLE]

On the other hand, for any $x,y\in X$, by (3.13) and (3.14), we have

[TABLE]

hence $({\mathcal{E}};V)$ is a linearisation of $\mathbf{k}$. We prove that it is minimal as well. To see this, note that for any $g\in{\mathbb{C}}^{X}_{0}$, with notation as in (3.12), we have

[TABLE]

hence, by (3.13), the linear span of $V(X)$ equals $Z^{X}_{K}$.

(2)$\Rightarrow$(1). This is proven exactly as in the classical case:

[TABLE]

for all $n\in{\mathbb{N}}$, $x_{1},\ldots,x_{n}\in X$, and $t_{1},\ldots,t_{n}\in{\mathcal{H}}$.

(3)$\Rightarrow$(2). Clear.

(1)$\Rightarrow$(3). Assuming that $Z$ is an admissible space, let $\mathbf{k}$ be positive semidefinite, let $({\mathcal{E}};V)$ be the weak VE-space linearisation of $\mathbf{k}$. Then, ${\mathcal{E}}$ is naturally equipped with a Hausdorff locally convex topology, see Subsection 2.3, and then completed to a VH- space ${\mathcal{H}}$. Thus, $({\mathcal{H}};V)$ is a weak VH-space linearisation of $\mathbf{k}$ and it is easy to see that it is topologically minimal. The fact that this completion can be made within $Z^{X}$ will follow from Proposition 3.8. ∎

3.3. Reproducing Kernel Spaces

Let $Z$ be an ordered $*$-space and let $X$ be a nonempty set. As in Subsection 3.1, we consider the complex vector space $Z^{X}$ of all functions $f\colon X\rightarrow Z$. A VE-space ${\mathcal{R}}$ over the ordered $*$-space $Z$ is called a weak $Z$-reproducing kernel VE-space on $X$ if there exists a Hermitian kernel $\mathbf{k}\colon X\times X\rightarrow Z$ such that the following axioms are satisfied:

• (rk1)

${\mathcal{R}}$ is a subspace of $Z^{X}$, with all algebraic operations.

• (rk2)

For all $x\in X$, the $Z$-valued map $\mathbf{k}_{x}=\mathbf{k}(\cdot,x)\colon X\rightarrow Z$ belongs to ${\mathcal{R}}$.

• (rk3)

For all $f\in{\mathcal{R}}$ we have $f(x)=[\mathbf{k}_{x},f]_{\mathcal{R}}$, for all $x\in X$.

The axiom (rk3) is called the reproducing property and note that, as a consequence, we have

[TABLE]

A weak $Z$-reproducing kernel VE-space $\mathbf{k}$ on $X$ is called minimal if

• (rk4)

$\operatorname{Lin}\{\mathbf{k}_{x}\mid x\in X\}={\mathcal{R}}$.

If $Z$ is an admissible space, a weak $Z$-reproducing kernel VE-space ${\mathcal{R}}$ that is a VH-space is called a weak $Z$-reproducing kernel VH-space. Such an ${\mathcal{R}}$ is called topologically minimal if

• (rk4)′

$\operatorname{Lin}\{\mathbf{k}_{x}\mid x\in X\}$ is dense in ${\mathcal{R}}$.

Remark 3.5**.**

Let ${\mathcal{R}}$ be a weak $Z$-reproducing kernel VH-space with respect to some admissible space $Z$. In general, the closed subspace $\overline{\operatorname{Lin}\{\mathbf{k}_{x}\mid x\in X\}}\subseteq{\mathcal{R}}$ may or may not be orthocomplemented in ${\mathcal{R}}$, see Subsection 2.3. This anomaly makes some differences when compared with the classical theory of reproducing kernel spaces, as is the case in closely related situations as in [3] and [4] as well.

Proposition 3.6**.**

A weak $Z$-reproducing kernel VH-space ${\mathcal{R}}$ with respect to some admissible space $Z$ is topologically minimal if and only if the closed subspace $\overline{\operatorname{Lin}\{\mathbf{k}_{x}\mid x\in X\}}$ is orthocomplemented in ${\mathcal{R}}$.

Proof.

If ${\mathcal{M}}:=\overline{\operatorname{Lin}\{\mathbf{k}_{x}\mid x\in X\}}$ is orthocomplemented then, as a consequence of (rk3), ${\mathcal{R}}$ is topologically minimal, in the sense of (rk4)′. Indeed, let $f\in{\mathcal{R}}$ be arbitrary. Since ${\mathcal{M}}$ is orthocomplemented, there exists $f_{1}\in{\mathcal{M}}$ and $f_{2}\in{\mathcal{M}}^{\bot}$ with $f=f_{1}+f_{2}$. By (rk3) we obtain that $0=[\mathbf{k}_{x},f_{2}]=f_{2}(x)$ for all $x\in{\mathcal{R}}$, and that $f_{2}=0$. It follows that $f\in{\mathcal{M}}$ and ${\mathcal{M}}={\mathcal{R}}$, i.e. $\operatorname{Lin}\{\mathbf{k}_{x}\mid x\in X\}$ is dense in ${\mathcal{R}}$. The converse implication is trivial. ∎

We first consider the relation between weak $Z$-reproducing kernel VE/VH-spaces and their reproducing kernels.

Proposition 3.7**.**

(a)* Let ${\mathcal{R}}$ be a weak $Z$-reproducing kernel VE-space on $X$, with respect to some ordered $*$-space $Z$ and with kernel $\mathbf{k}$. Then:*

• (i)

$\mathbf{k}$* is positive semidefinite and uniquely determined by ${\mathcal{R}}$.*

• (ii)

${\mathcal{R}}_{0}=\operatorname{Lin}\{\mathbf{k}_{x}\mid x\in X\}\subseteq{\mathcal{R}}$* is a minimal weak $Z$-reproducing kernel VE-space on $X$ and uniquely determined by $\mathbf{k}$ with this property.*

• (iii)

The gramian $[\cdot,\cdot]_{\mathcal{R}}$ is uniquely determined by $\mathbf{k}$ on ${\mathcal{R}}_{0}$.

(b)* Assume that $Z$ is admissible and ${\mathcal{R}}$ is a weak $Z$-reproducing kernel VH-space. Then:*

• (i)

$\overline{\mathcal{R}}_{0}$* is a topologically minimal $Z$-reproducing kernel VH-space in ${\mathcal{R}}$.*

• (ii)

The gramian $[\cdot,\cdot]_{\mathcal{R}}$ is uniquely determined by $\mathbf{k}$ on $\overline{\mathcal{R}}_{0}\subseteq{\mathcal{R}}$.

• (iii)

If ${\mathcal{R}}$ is topologically minimal then it is unique with this property.

Proof.

(a) Let $t_{1},\ldots,t_{n}\in{\mathbb{C}}$ and $x_{1},\ldots,x_{n}\in X$ be arbitrary. Using (3.15) it follows

[TABLE]

hence $\mathbf{k}$ is positive semidefinite. On the other hand, by (rk3) it follows that for all $x\in X$ the functions $\mathbf{k}_{x}$ are uniquely determined by $({\mathcal{R}};[\cdot,\cdot]_{\mathcal{R}})$, hence $\mathbf{k}(y,x)=\mathbf{k}_{x}(y)$, $x,y\in X$, are uniquely determined. Hence assertion (i) is proven. Assertion (ii) is clear by inspecting the definitions. Assertion (iii) is now clear by (rk3), see (3.15).

(b) The subspace $\overline{\mathcal{R}}_{0}$ of ${\mathcal{R}}$ is a topologically minimal $Z$-reproducing kernel VH-space, by definition. Using the assertion at item (a).(ii) and the continuity of the gramian $[\cdot,\cdot]_{\mathcal{R}}$, it follows that it is uniquely determined by $\mathbf{k}$ on $\overline{\mathcal{R}}_{0}$.

Assume that ${\mathcal{R}}$ is topologically minimal and let ${\mathcal{R}}^{\prime}$ be another topologically minimal weak $Z$-reproducing kernel VH-space on $X$ with the same kernel $\mathbf{k}$. By axiom (rk2) and the property (rk4), ${\mathcal{R}}_{0}=\operatorname{Lin}\{\mathbf{k}_{x}\mid x\in X\}$ is a linear space that lies and is dense in both of ${\mathcal{R}}$ and ${\mathcal{R}}^{\prime}$. By axiom (rk3), the $Z$-valued inner products $[\cdot,\cdot]_{\mathcal{R}}$ and $[\cdot,\cdot]_{{\mathcal{R}}^{\prime}}$ coincide on ${\mathcal{R}}_{0}$ and then, due to the special way in which the topologies on VH-spaces are defined, see (2.9) and (2.10), it follows that ${\mathcal{R}}$ and ${\mathcal{R}}^{\prime}$ induce the same topology on ${\mathcal{R}}_{0}$ hence, taking into account the density of ${\mathcal{R}}_{0}$ in both ${\mathcal{R}}$ and ${\mathcal{R}}^{\prime}$, we actually have ${\mathcal{R}}={\mathcal{R}}^{\prime}$ as VH-spaces. ∎

Consequently, given ${\mathcal{R}}$ a weak $Z$-reproducing kernel VE-space on $X$, without any ambiguity we can talk about the $Z$-reproducing kernel $\mathbf{k}$ corresponding to ${\mathcal{R}}$.

As a consequence of Proposition 3.7, weakly positive semidefiniteness is an intrinsic property of the reproducing kernel of any weak reproducing kernel VE-space. In the following we clarify in an explicit fashion the relation between weak VE/VH-linearisations and weak reproducing kernel VE/VH-spaces associated to positive semidefinite kernels.

Proposition 3.8**.**

Let $\mathbf{k}$ be a weakly positive semidefinite kernel on $X$ and with values in the ordered $*$-space $Z$.

(a)* Any weak reproducing kernel VE-space ${\mathcal{R}}$ associated to $\mathbf{k}$ gives rise to a weak VE-space linearisation $({\mathcal{E}};V)$ of $\mathbf{k}$, where ${\mathcal{E}}={\mathcal{R}}$ and*

[TABLE]

If ${\mathcal{R}}$ is minimal, then $({\mathcal{E}};V)$ is minimal.

(b)* Any minimal weak VE-space linearisation $({\mathcal{E}};V)$ of $\mathbf{k}$ gives rise to the minimal weak reproducing kernel VE-space ${\mathcal{R}}$, where*

[TABLE]

that is, ${\mathcal{R}}$ consists of all functions $X\ni x\mapsto[V(x),e]_{\mathcal{K}}\in Z$, for all $e\in{\mathcal{E}}$, in particular, ${\mathcal{R}}\subseteq Z^{X}$ and ${\mathcal{R}}$ is endowed with the algebraic operations inherited from the complex vector space $Z^{X}$.

Proof.

(a) Assume that $({\mathcal{R}};[\cdot,\cdot]_{\mathcal{R}})$ is a weak $Z$-reproducing kernel VE-space on $X$, with reproducing kernel $\mathbf{k}$. We let ${\mathcal{E}}={\mathcal{R}}$ and define $V$ as in (3.16). Note that $V(x)\in{\mathcal{E}}$ for all $x\in X$. Also, by (3.15) we have

[TABLE]

Thus, $({\mathcal{E}};V)$ is a weak VE-space linearisation of $\mathbf{k}$.

(b) Let $({\mathcal{E}};V)$ be a minimal weak VE-space linearisation of $\mathbf{k}$. Let ${\mathcal{R}}$ be defined by (3.17), that is, ${\mathcal{R}}$ consists of all functions $X\ni x\mapsto[V(x),h]_{\mathcal{E}}\in Z$, in particular ${\mathcal{R}}\subseteq Z^{X}$ with all algebraic operations inherited from the complex vector space $Z^{X}$.

The correspondence

[TABLE]

is clearly surjective. In order to verify that it is injective as well, let $h,g\in{\mathcal{E}}$ be such that $[V(\cdot),h]_{\mathcal{E}}=[V(\cdot),g]_{\mathcal{E}}$. Then, for all $x\in X$ we have

[TABLE]

equivalently,

[TABLE]

By the minimality of the linearisation $({\mathcal{E}};V)$ it follows that $g=h$. Thus, $U$ is a bijection.

Clearly, the bijective map $U$ defined at (3.18) is linear, hence a linear isomorphism of complex vector spaces ${\mathcal{E}}\rightarrow{\mathcal{R}}$. On ${\mathcal{R}}$ we introduce a $Z$-valued pairing

[TABLE]

Since $({\mathcal{E}};[\cdot,\cdot]_{\mathcal{E}})$ is a VE-space over $Z$, it follows that $({\mathcal{R}};[\cdot,\cdot]_{\mathcal{R}})$ is a VE-space over $Z$. Indeed, this follows from the observation that, by (3.20), we transported the $Z$-gramian from ${\mathcal{E}}$ to ${\mathcal{R}}$ or, in other words, we have defined on ${\mathcal{R}}$ the $Z$-gramian that makes the linear isomorphism $U$ a unitary operator between the VE-spaces ${\mathcal{E}}$ and ${\mathcal{R}}$.

We show that $({\mathcal{R}};[\cdot,\cdot]_{\mathcal{R}})$ is a weak $Z$-reproducing kernel VE-space with corresponding reproducing kernel $\mathbf{k}$. By definition, ${\mathcal{R}}\subseteq Z^{X}$. On the other hand, since

[TABLE]

taking into account that $V(x)\in{\mathcal{E}}$, by (3.17) it follows that $\mathbf{k}_{x}\in{\mathcal{R}}$ for all $x\in X$. Further, for all $f\in{\mathcal{R}}$ and all $x\in X$ we have

[TABLE]

where $g\in{\mathcal{E}}$ is the unique vector such that $[V(\cdot),g]_{\mathcal{E}}=f$, which shows that ${\mathcal{R}}$ satisfies the reproducing axiom as well. Finally, taking into account the minimality of the linearisation $({\mathcal{E}};V)$ and the definition (3.17), it follows that $\operatorname{Lin}\{\mathbf{k}_{x}\mid x\in X\}={\mathcal{R}}$. Thus, $({\mathcal{R}};[\cdot,\cdot]_{\mathcal{R}})$ is a minimal weak $Z$-reproducing kernel VE-space with reproducing kernel $\mathbf{k}$. ∎

Proposition 3.9**.**

Let $\mathbf{k}$ be a weakly positive semidefinite kernel on $X$ and valued in the admissible space $Z$.

(a)* Any weak reproducing kernel VH-space ${\mathcal{R}}$ associated to $\mathbf{k}$ gives rise to a weak VH-space linearisation $({\mathcal{H}};V)$ of $\mathbf{k}$, where ${\mathcal{H}}={\mathcal{R}}$ and*

[TABLE]

If ${\mathcal{R}}$ is topologically minimal then $({\mathcal{H}};V)$ is topologically minimal.

(b)* Any topologically minimal weak VH-space linearisation $({\mathcal{H}};V)$ of $\mathbf{k}$ gives rise to the topologically minimal weak reproducing kernel VH-space ${\mathcal{R}}$, where*

[TABLE]

that is, ${\mathcal{R}}$ consists of all functions $X\ni x\mapsto[V(x),e]_{\mathcal{K}}\in Z$, for all $e\in{\mathcal{H}}$, in particular, ${\mathcal{R}}\subseteq Z^{X}$ and ${\mathcal{R}}$ is endowed with the algebraic operations inherited from the complex vector space $Z^{X}$.

Proof.

(a) The argument is similar to that used to prove assertion (a) of Proposition 3.8.

(b) Let $({\mathcal{H}};V)$ be a topologically minimal weak VH-space linearisation of $\mathbf{k}$ and let ${\mathcal{R}}$ be defined as in (3.22). The correspondence

[TABLE]

is a linear bijection $U\colon{\mathcal{H}}\rightarrow{\mathcal{R}}$. The argument to support this claim is similar with that used during the proof of item (b) in Proposition 3.8, with the difference that from (3.19) we the topological minimality of the linearisation $({\mathcal{H}};V)$ in order to conclude that $g=h$. Thus, $U$ is a bijection.

On ${\mathcal{R}}$ we introduce a $Z$-valued pairing as in (3.20) Since $({\mathcal{H}};[\cdot,\cdot]_{\mathcal{H}})$ is a VH-space over $Z$, it follows that $({\mathcal{R}};[\cdot,\cdot]_{\mathcal{R}})$ is a VH-space over $Z$. This follows from the observation that, by (3.20), we transported the $Z$-gramian from ${\mathcal{H}}$ to ${\mathcal{R}}$ or, in other words, we have defined on ${\mathcal{R}}$ the $Z$-gramian that makes the linear isomorphism $U$ a unitary operator between the VH-spaces ${\mathcal{H}}$ and ${\mathcal{R}}$.

Finally, $({\mathcal{R}};[\cdot,\cdot]_{\mathcal{R}})$ is the topologically minimal weak $Z$-reproducing kernel VH-space with corresponding reproducing kernel $\mathbf{k}$. The argument is again similar with that used in the proof of item (b) in Proposition 3.8, with the difference that here we use the topological minimality. ∎

The following theorem adds one more characterisation of positive semidefinite kernels, when compared to Theorem 3.4, in terms of reproducing kernel spaces. It’s proof is a direct consequence of Proposition 3.8, Proposition 3.9, and Theorem 3.4.

Theorem 3.10**.**

(a)* Let $Z$ be an ordered $*$-space, $X$ a nonempty set, and $\mathbf{k}\colon X\times X\rightarrow Z$ a Hermitian kernel. The following assertions are equivalent:*

• (1)

$\mathbf{k}$* is weakly positive semidefinite.*

• (2)

$\mathbf{k}$* is the $Z$-valued reproducing kernel of a VE-space ${\mathcal{R}}$ in $Z^{X}$.*

(b)* If, in addition, $Z$ is an admissible space then assertions (1) and (2) are equivalent with*

• (3)

$\mathbf{k}$* is the $Z$-valued reproducing kernel of a VH-space ${\mathcal{R}}$ in $Z^{X}$.*

In particular, any weakly positive semidefinite $Z$-valued kernel $\mathbf{k}$ has a topologically minimal weak $Z$-reproducing kernel VH-space ${\mathcal{R}}$, uniquely determined by $\mathbf{k}$.

As a consequence of the last assertion of Theorem 3.10, given $\mathbf{k}\colon X\times X\rightarrow Z$ a positive semidefinite kernel for an admissible space $Z$, we can denote, without any ambiguity, by ${\mathcal{R}}_{\mathbf{k}}$ the unique topologically minimal weak $Z$-reproducing kernel VH-space on $X$ associated to $\mathbf{k}$.

4. Invariant Weakly Positive Semidefinite Kernels

Let $X$ be a nonempty set, a (multiplicative) semigroup $\Gamma$, and an action of $\Gamma$ on $X$, denoted by $\xi\cdot x$, for all $\xi\in\Gamma$ and all $x\in X$. By definition, we have

[TABLE]

This means that we have a semigroup morphism $\Gamma\ni\xi\mapsto\xi\cdot\in G(X)$, where $G(X)$ denotes the semigroup, with respect to composition, of all maps $X\rightarrow X$. In case the semigroup $\Gamma$ has a unit $\epsilon$, the action is called unital if $\epsilon\cdot x=x$ for all $x\in X$, equivalently, $\epsilon\cdot=\mathrm{Id}_{X}$.

We assume further that $\Gamma$ is a $*$-semigroup, that is, there is an involution $*$ on $\Gamma$; this means that $(\xi\eta)^{*}=\eta^{*}\xi^{*}$ and $(\xi^{*})^{*}=\xi$ for all $\xi,\eta\in\Gamma$. Note that, in case $\Gamma$ has a unit $\epsilon$ then $\epsilon^{*}=\epsilon$.

4.1. Invariant Weak VE-Space Linearisations.

Given an ordered $*$-space $Z$ we are interested in those Hermitian kernels $\mathbf{k}\colon X\times X\rightarrow Z$ that are invariant under the action of $\Gamma$ on $X$, that is,

[TABLE]

A triple $({\mathcal{E}};\pi;V)$ is called a invariant weak VE-space linearisation of the $Z$-valued kernel $\mathbf{k}$ and the action of $\Gamma$ on $X$, if:

• (ivel1)

$({\mathcal{E}};V)$ is a weak VE-space linearisation of the kernel $\mathbf{k}$.

• (ivel2)

$\pi\colon\Gamma\rightarrow{\mathcal{L}}^{*}({\mathcal{E}})$ is a $*$-representation, that is, a multiplicative $*$-morphism.

• (ivel3)

$V$ and $\pi$ are related by the formula: $V(\xi\cdot x)=\pi(\xi)V(x)$, for all $x\in X$, $\xi\in\Gamma$.

Let $({\mathcal{E}};\pi;V)$ be an invariant weak VE-space linearisation of the kernel $\mathbf{k}$. Since $({\mathcal{E}};V)$ is a weak linearisation and taking into account the axiom (ivel3), for all $x,y\in X$ and all $\xi\in\Gamma$, we have

[TABLE]

hence $\mathbf{k}$ is invariant under the action of $\Gamma$ on $X$.

If, in addition to the axioms (ivel1)–(ivel3), the triple $({\mathcal{E}};\pi;V)$ has also the property

• (ivel4)

$\operatorname{Lin}V(X)={\mathcal{E}}$,

that is, the linearisation $({\mathcal{E}};V)$ is minimal, then $({\mathcal{E}};\pi;V)$ is called minimal.

Remarks 4.1**.**

(a) The minimality condition (ivel4) does not depend on the representation $\pi$. Apparently, $\operatorname{Lin}\pi(\Gamma)V(X)$ looks like a suitable candidate to replace $\operatorname{Lin}V(X)$ in (ivel4). However, in case the $*$-semigroup has a unit, then we have $\operatorname{Lin}\pi(\Gamma)V(X)=\operatorname{Lin}V(X)$ but, when $\Gamma$ does not have a unit, only the inclusion $\operatorname{Lin}\pi(\Gamma)V(X)\subseteq\operatorname{Lin}V(X)$ holds and hence, $\operatorname{Lin}\pi(\Gamma)V(X)$ may be too small to accommodate all $V(x)$ for $x\in X$.

(b) Let $({\mathcal{E}};\pi;V)$ be an invariant weak VE-space linearisation of the kernel $\mathbf{k}$. Then, for each $\gamma\in\Gamma$, we $\pi(\gamma)V(x)=V(\gamma\cdot x)$, for all $x\in X$, hence $\pi(\gamma)$ leaves invariant $\operatorname{Lin}V(X)$ and, consequently, letting ${\mathcal{E}}_{0}=\operatorname{Lin}V(X)$, $\pi_{0}\colon\Gamma\rightarrow{\mathcal{L}}^{*}({\mathcal{E}}_{0})$ defined by $\pi_{0}(\gamma)f=\pi(\gamma)f$ for all $f\in\operatorname{Lin}V(X)$, and $V_{0}\colon X\rightarrow{\mathcal{E}}_{0}$ defined by $V_{0}(x)=V(x)$, it follows that $({\mathcal{E}}_{0};\pi_{0};V_{0})$ is a minimal invariant weak VE-space linearisation of the kernel $\mathbf{k}$.

As usually [34], minimal invariant VE-space linearisations preserve linearity.

Proposition 4.2**.**

Assume that, given an ordered $*$-space $Z$ valued kernel $\mathbf{k}$, invariant under the action of the $*$-semigroup $\Gamma$ on $X$, for some fixed $\alpha,\beta,\gamma\in\Gamma$ we have $\mathbf{k}(y,\alpha\cdot x)+\mathbf{k}(y,\beta\cdot x)=\mathbf{k}(y,\gamma\cdot x)$ for all $x,y\in X$. Then for any minimal weak invariant VE-space linearisation $({\mathcal{E}};\pi;V)$ of $\mathbf{k}$, the representation satisfies $\pi(\alpha)+\pi(\beta)=\pi(\gamma)$.

Proof.

For any $x,y\in X$ we have

[TABLE]

hence, since $V(X)$ linearly spans ${\mathcal{E}}$, it follows that $\pi(\alpha)+\pi(\beta)=\pi(\gamma)$. ∎

Two invariant weak VE-space linearisations $({\mathcal{E}};\pi;V)$ and $({\mathcal{E}}^{\prime};\pi^{\prime};V^{\prime})$, of the same Hermitian kernel $\mathbf{k}$, are called unitarily equivalent if there exists a unitary operator $U\colon{\mathcal{E}}\rightarrow{\mathcal{E}}^{\prime}$ such that $U\pi(\xi)=\pi^{\prime}(\xi)U$ for all $\xi\in\Gamma$, and $UV(x)=V^{\prime}(x)$ for all $x\in X$. Let us note that, in case both of these invariant weak VE-space linearisations are minimal, then this is equivalent with the requirement that the weak VE-space linearisations $({\mathcal{E}};V)$ and $({\mathcal{E}}^{\prime};V^{\prime})$ are unitary equivalent.

Here have the first main theorem of this article in which invariant weakly positive semidefinite kernels are characterised by invariant weak VE-space linearisations and by certain $*$-representations on weak $Z$-reproducing kernel VE-spaces.

Theorem 4.3**.**

Let $\Gamma$ be a $*$-semigroup that acts on the nonempty set $X$, and let $\mathbf{k}\colon X\times X\rightarrow Z$ be a $Z$-valued kernel for some ordered $*$-space $Z$. The following assertions are equivalent:

• (1)

$\mathbf{k}$* satisfies the following conditions:*

• (a)

$\mathbf{k}$* is weakly positive semidefinite.*

• (b)

$\mathbf{k}$* is invariant under the action of $\Gamma$ on $X$, that is, (4.2) holds.*

• (2)

$\mathbf{k}$* has an invariant weak VE-space linearisation $({\mathcal{E}};\pi;V)$.*

• (3)

$\mathbf{k}$* admits a weak $Z$-reproducing kernel VE-space ${\mathcal{R}}$ and there exists a $*$-representation $\rho\colon\Gamma\rightarrow{\mathcal{L}}^{*}({\mathcal{R}})$ such that $\rho(\xi)\mathbf{k}_{x}=\mathbf{k}_{\xi\cdot x}$ for all $\xi\in\Gamma$, $x\in X$.*

Moreover, in case any of the assertions (1), (2), or (3) holds, then a minimal invariant weak VE-space linearisation of $\mathbf{k}$ can be constructed and a minimal weak $Z$-reproducing kernel ${\mathcal{R}}$ as in (3) can constructed as well.

Proof.

(1)$\Rightarrow$(2). We consider the notation and the minimal weak VE-space linearisation $({\mathcal{E}};V)$ constructed as in the proof of the implication (1)$\Rightarrow$(2) of Theorem 3.4. For each $\xi\in\Gamma$ we let $\pi(\xi)\colon Z^{X}\rightarrow Z^{X}$ be defined by

[TABLE]

We claim that $\pi(\xi)$ leaves $Z^{X}_{K}$ invariant, where $K$ is the convolution operator defined at (3.3) and $Z^{X}_{K}\subseteq Z^{X}$ denotes its range. To see this, let $f\in Z^{X}_{K}$, that is, $f=Kg$ for some $g\in{\mathbb{C}}^{X}_{0}$ or, even more explicitly, by (3.10),

[TABLE]

Then,

[TABLE]

where,

[TABLE]

Since clearly $g_{\xi}\in{\mathbb{C}}^{X}_{0}$, that is, $g_{\xi}$ has finite support, it follows that $\pi(\xi)$ leaves $Z^{X}_{K}$ invariant. In the following we denote by the same symbol $\pi(\xi)$ the map $\pi(\xi)\colon Z^{X}_{K}\rightarrow Z^{X}_{K}$.

In the following we prove that $\pi$ is a representation of the semigroup $\Gamma$ on the complex vector space $Z^{X}_{K}$, that is,

[TABLE]

To see this, let $f\in Z^{X}_{K}$ be fixed and denote $h=\pi(\beta)f$, that is, $h(y)=f(\beta^{*}\cdot y)$ for all $y\in X$. Then $\pi(\alpha)\pi(\beta)f=\pi(\alpha)h$, that is, $(\pi(\alpha)h)(y)=h(\alpha^{*}\cdot y)=f(\beta^{*}\alpha^{*}\cdot y)=f((\alpha\beta)^{*}\cdot y)=(\pi(\alpha\beta))(y)$, for all $y\in X$, which proves (4.7).

Next we show that $\pi$ is actually a $*$-representation, that is,

[TABLE]

To see this, let $f=Kg$ and $f^{\prime}=Kg^{\prime}$ for some $g,g^{\prime}\in{\mathbb{C}}^{X}_{0}$. Then, by (3.11) and (4.6),

[TABLE]

and hence the formula (4.8) is proven.

In order to show that the axiom (vel3) holds as well, we use (3.14) and (4.4). Thus, for all $\xi\in\Gamma$, $x,y\in X$ and taking into account that $\mathbf{k}$ is invariant under the action of $\Gamma$ on $X$, we have

[TABLE]

which proves (vel3). Thus, $({\mathcal{E}};\pi;V)$, here constructed, is an invariant weak VE-space linearisation of the Hermitian kernel $\mathbf{k}$. Note that $({\mathcal{E}};\pi;V)$ is minimal, that is, the axiom (vel4) holds, since the linearisation $({\mathcal{E}};V)$ is minimal, by the proof of Theorem 3.4.

In order to prove the uniqueness of the minimal weak invariant linearisation, let $({\mathcal{K}}^{\prime};\pi^{\prime};V^{\prime})$ be another minimal invariant weak VE-space linearisation of $\mathbf{k}$. We consider the unitary operator $U\colon{\mathcal{K}}\rightarrow{\mathcal{K}}^{\prime}$ defined as in (3.9) and we already know that $UV(x)=V^{\prime}(x)$ for all $x\in X$. Since, for any $\xi\in\Gamma$, $x\in X$ we have

[TABLE]

and taking into account the minimality, it follows that $U\pi(\xi)=\pi^{\prime}(\xi)U$ for all $\xi\in\Gamma$.

(2)$\Rightarrow$(1). Let $({\mathcal{E}};\pi;V)$ be an invariant weak VE-space linearisation of the kernel $\mathbf{k}$. We already know from the proof of Theorem 3.4 that $\mathbf{k}$ is positive semidefinite and it was shown in (4.3) that $\mathbf{k}$ is invariant under the action of $\Gamma$ on $X$.

(2)$\Rightarrow$(3). Let $({\mathcal{E}};\pi;V)$ be an invariant weak VE-space linearisation of the kernel $\mathbf{k}$ and the action of $\Gamma$ on $X$. Without loss of generality, we can assume that it is minimal. Indeed, since we have already proven the implication (2)$\Rightarrow$(1), we observe that during the proof of the implication (1)$\Rightarrow$(2), we obtained a minimal invariant weak VE-space linearisation of $\mathbf{k}$.

We use the notation and the facts established during the proof of the implication (2)$\Rightarrow$(3) in Theorem 3.10. Then, for all $x,y\in X$ we have

[TABLE]

hence, letting $\rho(\xi)=U\pi(\xi)U^{-1}$, where $U\colon{\mathcal{K}}\rightarrow{\mathcal{R}}$ is the unitary operator defined as in (3.18), we obtain a $*$-representation of $\Gamma$ on the VE-space ${\mathcal{R}}$ such that $\mathbf{k}_{\xi\cdot x}=\rho(\xi)\mathbf{k}_{x}$ for all $\xi\in\Gamma$ and $x\in X$.

(3)$\Rightarrow$(2). Let $({\mathcal{R}};\rho)$, where ${\mathcal{R}}$ is a weak $Z$-reproducing kernel VE-space of $\mathbf{k}$ and $\rho\colon\Gamma\rightarrow{\mathcal{L}}^{*}({\mathcal{R}})$ is a $*$-representation such that $\mathbf{k}_{\xi\cdot x}=\rho(\xi)\mathbf{k}_{x}$ for all $\xi\in\Gamma$ and $x\in X$. As in the proof of the implication (3)$\Rightarrow$(2) in Theorem 3.10, we show that $({\mathcal{R}};V)$, where $V$ is defined as in (3.16), is a minimal linearisation of $\mathbf{k}$. Letting $\pi=\rho$, it is then easy to see that $({\mathcal{R}};\pi;V)$ is an invariant weak VE-space linearisation of the kernel $\mathbf{k}$ and the action of $\Gamma$ on $X$. ∎

4.2. Boundedly Adjointable Invariant Weak VH-Space Linearisations.

Let us assume now that $Z$ is an admissible space and $\mathbf{k}\colon X\times X\rightarrow Z$ is a kernel. A triple $({\mathcal{K}};\pi;V)$ is called a boundedly adjointable invariant weak VH-space linearisation of the $Z$-valued kernel $\mathbf{k}$ and the action of $\Gamma$ on $X$, if:

• (ivhl1)

$({\mathcal{K}};V)$ is a weak VH-space linearisation of the kernel $\mathbf{k}$.

• (ivhl2)

$\pi\colon\Gamma\rightarrow{\mathcal{B}}^{*}({\mathcal{K}})$ is a $*$-representation, that is, a multiplicative $*$-morphism.

• (ivhl3)

$V$ and $\pi$ are related by the formula: $V(\xi\cdot x)=\pi(\xi)V(x)$, for all $x\in X$, $\xi\in\Gamma$.

Let $({\mathcal{K}};\pi;V)$ be a boundedly adjointable invariant weak VH-space linearisation of the kernel $\mathbf{k}$. As in (4.3), it follows that $\mathbf{k}$ is invariant under the action of $\Gamma$ on $X$.

If, in addition to the axioms (ivhl1), (ivhl2), and (ivhl3), the triple $({\mathcal{K}};\pi;V)$ has also the property

• (ivhl4)

$\operatorname{Lin}V(X)$ is dense in ${\mathcal{K}}$,

that is, the weak VH-space linearisation $({\mathcal{H}};V)$ is topologically minimal, then $({\mathcal{K}};\pi;V)$ is called topologically minimal. A similar observation as in Remark 4.1 can be made: in case $\Gamma$ has a unit then (ivhl4) is equivalent with saying $\operatorname{Lin}\pi(\Gamma)V(X)$ is dense in ${\mathcal{K}}$ but, in general the apparently more candidate $\operatorname{Lin}\pi(\Gamma)V(X)$ is too small to provide a suitable topological minimality condition.

Two boundedly adjointable invariant weak VH-space linearisations $({\mathcal{K}};\pi;V)$ and $({\mathcal{K}}^{\prime};\pi^{\prime};V^{\prime})$ of the same kernel $\mathbf{k}$ are unitarily invariant if there exists a unitary $U\in{\mathcal{B}}^{*}({\mathcal{K}},{\mathcal{K}}^{\prime})$ such that $U\pi(\xi)=\pi^{\prime}(\xi)U$ for all $\xi\in\Gamma$ and $UV(x)=V^{\prime}(x)$ for all $x\in X$. Let us note that, in case both of these boundedly adjointable invariant weak VH-space linearisations are topologically minimal then they are unitarily equivalent.

The analog of Proposition 4.2 for topologically minimal invariant weak VH-space linearisations holds as well.

Proposition 4.4**.**

Assume that, given an admissible space $Z$ and a $Z$-valued kernel $\mathbf{k}$, invariant under the action of the $*$-semigroup $\Gamma$ on $X$, for some fixed $\alpha,\beta,\gamma\in\Gamma$ we have $\mathbf{k}(y,\alpha\cdot x)+\mathbf{k}(y,\beta\cdot x)=\mathbf{k}(y,\gamma\cdot x)$ for all $x,y\in X$. Then, for any topologically minimal boundedly adjointable invariant weak VH-space linearisation $({\mathcal{K}};\pi;V)$ of $\mathbf{k}$, the representation satisfies $\pi(\alpha)+\pi(\beta)=\pi(\gamma)$.

Proof.

The same argument as in the proof of Proposition 4.2 applies with the small difference that we use the topological minimality and get the same conclusion. ∎

Here we have the second main theorem of this article in which invariant weakly positive semidefinite kernels are characterised by boundedly adjointable invariant weak VE-space linearisations and by certain $*$-representations with boundedly adjointable operators on weak $Z$-reproducing kernel VE-spaces. This is the first topological analogue of Theorem 4.3.

Theorem 4.5**.**

Let $\Gamma$ be a $*$-semigroup that acts on the nonempty set $X$, and let $\mathbf{k}\colon X\times X\rightarrow Z$ be a $Z$-valued kernel for some admissible space $Z$. The following assertions are equivalent:

• (1)

$\mathbf{k}$* satisfies the following conditions:*

• (a)

$\mathbf{k}$* is weakly positive semidefinite.*

• (b)

$\mathbf{k}$* is invariant under the action of $\Gamma$ on $X$, that is, (4.2) holds.*

• (c)

For any $\alpha\in\Gamma$ there exists $c(\alpha)\geq 0$ such that

[TABLE]

for $n\in{\mathbb{N}}$, all $x_{1},\ldots,x_{n}\in X$, and all $t_{1},\ldots,t_{n}\in{\mathbb{C}}$.

• (2)

$\mathbf{k}$* has a boundedly adjointable invariant weak VH-space linearisation $({\mathcal{K}};\pi;V)$.*

• (3)

$\mathbf{k}$* admits a weak $Z$-reproducing kernel VH-space ${\mathcal{R}}$ and there exists a $*$-representation $\rho\colon\Gamma\rightarrow{\mathcal{B}}^{*}({\mathcal{R}})$ such that $\rho(\xi)\mathbf{k}_{x}=\mathbf{k}_{\xi\cdot x}$ for all $\xi\in\Gamma$, $x\in X$.*

Moreover, in case any of the assertions (1), (2), or (3) holds, then a topologically minimal boundedly adjointable invariant weak VH-space linearisation can be constructed and a topologically minimal weak $Z$-reproducing kernel VH-space ${\mathcal{R}}$ as in assertion (3) can be constructed as well.

Proof.

(1)$\Rightarrow$(2). We consider the notation and the minimal invariant weak VE-space linearisation $({\mathcal{E}};\pi;V)$ constructed as in the proof of the implication (1)$\Rightarrow$(2) of Theorem 4.3. Considering $Z^{X}_{K}$ as a VE-space with $Z$-gramian $[\cdot,\cdot]_{\mathcal{E}}$, we consider its natural topology as in Subsection 2.3 and we prove now that $\pi(\xi)$ is bounded for all $\xi\in\Gamma$. Indeed, let $f=Kg$ for some $g\in{\mathbb{C}}^{X}_{0}$. Using the definition of $\pi(\xi)$ and the boundedness condition (c), we have

[TABLE]

and hence the boundedness of $\pi(\xi)$ is proven. This implies that $\pi(\xi)$ can be uniquely extended by continuity to an operator $\pi(\xi)\in{\mathcal{B}}({\mathcal{K}})$. In addition, since $\pi(\xi^{*})$ also extends by continuity to an operator $\pi(\xi^{*})\in{\mathcal{B}}({\mathcal{K}})$ and taking into account (4.8), it follows that $\pi(\xi)$ is adjointable and $\pi(\xi^{*})=\pi(\xi)^{*}$. We conclude that $\pi$ is a $*$-representation of $\Gamma$ in ${\mathcal{B}}^{*}({\mathcal{K}})$, that is, the axiom (ivhl2) holds.

The uniqueness of the topologically minimal boundedly adjointable invariant weak VH-space linearisation follows as usually.

(2)$\Rightarrow$(1). Let $({\mathcal{K}};\pi;V)$ be a boundedly adjointable invariant weak VH-space linearisation of the kernel $\mathbf{k}$. We already know from the proof of Theorem 3.4 that $\mathbf{k}$ is positive semidefinite and it was shown in (4.3) that $\mathbf{k}$ is invariant under the action of $\Gamma$ on $X$. In order to show that the boundedness condition (c) holds as well, let $\alpha\in\Gamma$, $n\in{\mathbb{N}}$, $x_{1},\ldots,x_{n}\in X$, and $t_{1},\ldots,t_{n}\in{\mathbb{C}}$ be arbitrary. Then

[TABLE]

and hence (c) holds with $c(\alpha)=\|\pi(\alpha)\|\geq 0$.

(2)$\Rightarrow$(3). Let $({\mathcal{K}};\pi;V)$ be a boundedly adjointable invariant weak VH-space linearisation of the kernel $\mathbf{k}$ with respect to the action of $\Gamma$ on $X$. Without loss of generality we can assume that it is topologically minimal. Indeed, since we have already proven the implication (2)$\Rightarrow$(1), we observe that during the proof of the implication (1)$\Rightarrow$(2), we obtained a topologically minimal invariant weak VH-space linearisation of $\mathbf{k}$.

We use the notation and the facts established during the proof of the implication (2)$\Rightarrow$(3) in Theorem 3.10. Then, for all $x,y\in X$ we have

[TABLE]

hence, letting $\rho(\xi)=U\pi(\xi)U^{-1}$, where $U\colon{\mathcal{K}}\rightarrow{\mathcal{R}}$ is the unitary operator defined as in (3.18), we obtain a $*$-representation of $\Gamma$ on the VH-space ${\mathcal{R}}$ such that $\mathbf{k}_{\xi\cdot x}=\rho(\xi)\mathbf{k}_{x}$ for all $\xi\in\Gamma$ and $x\in X$.

(3)$\Rightarrow$(2). Let $({\mathcal{R}};\rho)$, where ${\mathcal{R}}={\mathcal{R}}(\mathbf{k})$ is the weak reproducing kernel VH-space of $\mathbf{k}$ and $\rho\colon\Gamma\rightarrow{\mathcal{B}}^{*}({\mathcal{R}})$ is a $*$-representation such that $\mathbf{k}_{\xi\cdot x}=\rho(\xi)\mathbf{k}_{x}$ for all $\xi\in\Gamma$ and $x\in X$. As in the proof of the implication (3)$\Rightarrow$(2) in Theorem 3.10, we show that $({\mathcal{R}};V)$, where $V$ is defined as in (3.16), is a minimal weak linearisation of $\mathbf{k}$. Letting $\pi=\rho$, it is then easy to see that $({\mathcal{R}};\pi;V)$ is a boundedly adjointable invariant weak VH-space linearisation of the kernel $\mathbf{k}$ with respect to the action of $\Gamma$ on $X$. ∎

4.3. Continuously Adjointable Invariant Weak VH-Space Linearisations.

Let $Z$ be an admissible space. A triple $({\mathcal{K}};\pi;V)$ is called a continuously adjointable invariant weak VH-space linearisation of the $Z$-valued kernel $\mathbf{k}$ and the action of $\Gamma$ on $X$, if the requirements (ivhl1) and (ivhl2) holds and, instead of (ihvl2), it satisfies

• (ivhl2)′

$\pi\colon\Gamma\rightarrow{\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{K}})$ is a $*$-representation, that is, a multiplicative $*$-morphism.

Clearly, for any continuously adjointable invariant weak VH-space linearisation $({\mathcal{K}};\pi;V)$ of the kernel $\mathbf{k}$, it follows that $\mathbf{k}$ is invariant under the action of $\Gamma$ on $X$.

If, in addition to the axioms (ivhl1), (ivhl2)′, and (ivhl3), the triple $({\mathcal{K}};\pi;V)$ has also the property (ivhl4), that is, the weak VH-space linearisation $({\mathcal{H}};V)$ is topologically minimal, then $({\mathcal{K}};\pi;V)$ is called a topologically minimal continuously adjointable invariant weak VH-space linearisation of the $Z$-kernel $\mathbf{k}$ with respect to the action of $\Gamma$ on $X$.

The unitary equivalence of two continuously adjointable invariant weak VH-space linearisations $({\mathcal{K}};\pi;V)$ and $({\mathcal{K}}^{\prime};\pi^{\prime};V^{\prime})$ of the same kernel $\mathbf{k}$ is defined as in the case of boundedly adjointable invariant weak VH-space linearisations and their topological minimality implies their unitary equivalence.

The analog of Proposition 4.2 for topologically minimal continuously adjointable invariant weak VH-space linearisations holds as well.

The next theorem is the analogue of Theorem 4.5 for continuously adjointable invariant weak VH-space linearisations in which the boundedness condition $1.(c)$ of Theorem 4.5 is replaced with a weaker one.

Theorem 4.6**.**

Let $\Gamma$ be a $*$-semigroup that acts on the nonempty set $X$, and let $\mathbf{k}\colon X\times X\rightarrow Z$ be a $Z$-valued kernel for some admissible space $Z$. The following assertions are equivalent:

• (1)

$\mathbf{k}$* satisfies the following conditions:*

• (a)

$\mathbf{k}$* is weakly positive semidefinite.*

• (b)

$\mathbf{k}$* is invariant under the action of  $\Gamma$ on $X$, that is, (4.2) holds.*

• (c)

For any $\alpha\in\Gamma$ and any seminorm $p\in S(Z)$, there exists a seminorm $q\in S(Z)$ and a constant $c(\alpha)\geq 0$ such that

[TABLE]

for $n\in{\mathbb{N}}$, all $x_{1},\ldots,x_{n}\in X$, and all $t_{1},\ldots,t_{n}\in{\mathbb{C}}$.

• (2)

$\mathbf{k}$* has a continuously adjointable invariant weak VH-space linearisation $({\mathcal{K}};\pi;V)$.*

• (3)

$\mathbf{k}$* admits a weak $Z$-reproducing kernel VH-space ${\mathcal{R}}$ and there exists a $*$-representation $\rho\colon\Gamma\rightarrow{\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{R}})$ such that $\rho(\xi)\mathbf{k}_{x}=\mathbf{k}_{\xi\cdot x}$ for all $\xi\in\Gamma$, $x\in X$.*

Moreover, in case any of the assertions (1), (2), or (3) holds, then a topologically minimal continuously adjointable invariant VH-space linearisation can be constructed and a topologically minimal weak $Z$-reproducing kernel VH-space ${\mathcal{R}}$ as in assertion (3) can be constructed as well.

Proof.

(1)$\Rightarrow$(2). We consider the notation and constructions as in the proof of the implication (1)$\Rightarrow$(2) of Theorem 4.3, and follow the same idea as in the proof of the implication (1)$\Rightarrow$(2) of Theorem 4.5, with the difference that the weak boundedness condition 1.(c) is used. For any $\xi\in\Gamma$, $f=Kg$ and $p\in S(Z)$ there exist $q\in S(Z)$ and $c(\xi)\geq 0$ such that

[TABLE]

hence the continuity of $\pi(\xi)$ is proven. This implies that $\pi(\xi)$ can be uniquely extended by continuity to an operator $\pi(\xi)\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{K}})$. In addition, since $\pi(\xi^{*})$ also extends by continuity to an operator $\pi(\xi^{*})\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{K}})$ and taking into account (4.8), it follows that $\pi(\xi)$ is adjointable and $\pi(\xi^{*})=\pi(\xi)^{*}$. We conclude that $\pi$ is a $*$-representation of $\Gamma$ in ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{K}})$.

The uniqueness of the topologically minimal continuously adjointable invariant VH-space linearisation follows as usually.

(2)$\Rightarrow$(1). By the proof of the implication (2)$\Rightarrow$(1) of Theorem 4.5, we only have to show that the boundedness condition (c) holds. Let $\alpha\in\Gamma$, $n\in{\mathbb{N}}$, $x_{1},\ldots,x_{n}\in X$, and $t_{1},\ldots,t_{n}\in{\mathbb{C}}$ be arbitrary. Then, due to the continuity of $\pi(\alpha)$ and taking into account the $S(Z)$ is directed, there exist $q\in S(Z)$ and $c(\alpha)\geq 0$ such that

[TABLE]

(2)$\Rightarrow$(3). Let $({\mathcal{K}};V;\pi)$ be a continuously adjointable weak VH-space linearisation of the kernel $\mathbf{k}$ with respect to the action of $\Gamma$ on $X$. Using exactly the same ideas in the proof of the implication (2)$\Rightarrow$(1) of Theorem 4.5, we obtain a continuous $*$-representation of $\Gamma$ on the VH-space ${\mathcal{R}}$ defined by $\rho(\xi)=U\pi(\xi)U^{-1}$, where $U\colon{\mathcal{K}}\rightarrow{\mathcal{R}}$ is the unitary operator defined as in (3.18).

(3)$\Rightarrow$(2). Let $({\mathcal{R}};\rho)$, where ${\mathcal{R}}={\mathcal{R}}(\mathbf{k})$ is the weak reproducing kernel VH-space of $\mathbf{k}$ and $\rho\colon\Gamma\rightarrow{\mathcal{L}}_{\mathrm{c}}^{*}({\mathcal{R}})$ is a $*$-representation such that $\mathbf{k}_{\xi\cdot x}=\rho(\xi)\mathbf{k}_{x}$ for all $\xi\in\Gamma$ and $x\in X$. As in the proof of the implication (3)$\Rightarrow$(2) in Theorem 4.5, letting $\pi=\rho$, it is then easy to see that $({\mathcal{R}};\pi;V)$ is a weak VH-space linearisation of the kernel $\mathbf{k}$ and $\pi$ satisfies the required properties. ∎

5. Unification of Some Dilation Theorems

In this section we show how various dilation theorems can be obtained as special cases of Theorem 4.3, Theorem 4.5, and Theorem 4.6.

5.1. Invariant Kernels with Values Adjointable Operators.

We show that Theorem 2.8 in [3] can be seen as a special case of Theorem 4.3. We first recall necessary definitions from [3].

In this subsection we will consider a kernel on a nonempty set $X$ and taking values in ${\mathcal{L}}^{*}({\mathcal{H}})$, for a VE-space ${\mathcal{H}}$ over an ordered $*$-space $Z$, that is, a map $\mathbf{l}\colon X\times X\rightarrow{\mathcal{L}}^{*}({\mathcal{H}})$.

A kernel $\mathbf{l}\colon X\times X\rightarrow{\mathcal{L}}^{*}({\mathcal{H}})$ is called positive semidefinite if for all $n\in{\mathbb{N}}$, $x_{1},x_{2},\cdots,x_{n}\in X$, and $h_{1},h_{2},\cdots,h_{n}\in{\mathcal{H}}$, we have

[TABLE]

An invariant ${\mathcal{L}}^{*}({\mathcal{H}})$-valued VE-space linearisation of a kernel $\mathbf{l}$ and an action of a $*$-semigroup $\Gamma$ on $X$ is, by definition, a triple $(\tilde{{\mathcal{E}}};\tilde{\pi};\tilde{V})$ such that

• (hvel1)

$\tilde{{\mathcal{E}}}$ is a VE-space over the same ordered $*$-space $Z$,

• (hvel2)

$\tilde{\pi}\colon\Gamma\rightarrow{\mathcal{L}}^{*}(\tilde{\mathcal{E}})$ is a $*$-representation,

• (hvel3)

$\tilde{V}\colon X\rightarrow{\mathcal{L}}^{*}({\mathcal{H}},\tilde{{\mathcal{E}}})$, satisfying $\mathbf{k}(x,y)=\tilde{V}(x)^{*}\tilde{V}(y)$ for all $x,y\in X$ and $\tilde{V}(\xi\cdot x)=\tilde{\pi}(\xi)\tilde{V}(x)$ for all $x\in X$, $\xi\in\Gamma$.

If an invariant ${\mathcal{L}}^{*}({\mathcal{H}})$-valued VE-space linearisation has the property that $\operatorname{Lin}V(X){\mathcal{H}}=\tilde{{\mathcal{E}}}$, then it is called minimal. Two invariant ${\mathcal{L}}^{*}({\mathcal{H}})$-VE-space linearisations $(\tilde{{\mathcal{E}}};\tilde{\pi};\tilde{V})$ and $(\tilde{{\mathcal{F}}};\tilde{\rho};\tilde{W})$ of the same kernel $\mathbf{l}$ are called unitarily equivalent if there exists a unitary operator $U:\tilde{{\mathcal{E}}}\rightarrow\tilde{{\mathcal{F}}}$ such that $U\tilde{\pi}(\gamma)=\tilde{\rho}(\gamma)U$ for all $\gamma\in\Gamma$ and $U\tilde{V}(x)=\tilde{W}(x)$ for all $x\in X$.

Let ${\mathcal{H}}^{X}$ be the vector space of all maps $f\colon X\rightarrow{\mathcal{H}}$, for a nonempty set $X$ and a VE-space ${\mathcal{H}}$ over the ordered $*$-space $Z$. A VE-space $\tilde{{\mathcal{R}}}$ over the same ordered $*$-space $Z$ is called a ${\mathcal{L}}^{*}({\mathcal{H}})$-reproducing kernel VE-space on $X$ of the kernel $\mathbf{l}$ if

• (hrk1)

$\tilde{{\mathcal{R}}}$ is a vector subspace of ${\mathcal{H}}^{X}$.

• (hrk2)

For all $x\in X$ and $h\in{\mathcal{H}}$, the ${\mathcal{H}}$-valued function $\mathbf{l}_{x}h:=\mathbf{l}(\cdot,x)h$ belongs to $\tilde{{\mathcal{R}}}$.

• (hrk3)

For all $f\in\tilde{{\mathcal{R}}}$ we have $[f(x),h]_{{\mathcal{H}}}=[f,\mathbf{l}_{x}h]_{\tilde{{\mathcal{R}}}}$ for all $x\in X$ and $h\in{\mathcal{H}}$.

The space $\tilde{{\mathcal{R}}}$ is minimal if $\tilde{{\mathcal{R}}}=\operatorname{Lin}\{\mathbf{l}_{x}h\mid x\in X,\ h\in{\mathcal{H}}\}$.

Theorem 5.1** (Theorem 2.8 in [3]).**

Let $\Gamma$ be a $*$-semigroup acting on a nonempty set $X$, ${\mathcal{H}}$ be a VE-space on an ordered $*$-space $Z$, and $\mathbf{l}\colon X\times X\rightarrow{\mathcal{L}}^{*}({\mathcal{H}})$ be a kernel. The following assertions are equivalent:

• (1)

$\mathbf{l}$* satisfies the following properties:*

• (a)

$\mathbf{l}$* is positive semidefinite.*

• (b)

$\mathbf{l}$* is invariant under the action of $\Gamma$ on $X$.*

• (2)

$\mathbf{l}$* has an invariant ${\mathcal{L}}^{*}({\mathcal{H}})$-valued VE-space linearisation $(\tilde{{\mathcal{E}}};\tilde{\pi};\tilde{V})$.*

• (3)

$\mathbf{l}$* admits a ${\mathcal{L}}^{*}({\mathcal{H}})$-reproducing kernel VE-space $\tilde{{\mathcal{R}}}$ and there exists a $*$-representation $\tilde{\rho}:\Gamma\rightarrow{\mathcal{L}}^{*}(\tilde{{\mathcal{R}}})$ such that $\tilde{\rho}(\xi)\mathbf{l}_{x}h=\mathbf{l}_{\xi\cdot x}h$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$.*

Moreover, in case any of the assertions (1), (2) or (3) holds, a minimal invariant ${\mathcal{L}}^{*}({\mathcal{H}})$-VE-space linearisation can be constructed, and a pair $(\tilde{{\mathcal{R}}};\tilde{\rho})$ as in (3) with $\tilde{{\mathcal{R}}}$ can be always obtained as well.

Proof.

(1)$\Rightarrow$(2). Define a kernel $\mathbf{k}\colon(X\times{\mathcal{H}})\times(X\times{\mathcal{H}})\rightarrow Z$ by

[TABLE]

Since $\mathbf{l}$ is semipositive definite in the sense of (5.1), $\mathbf{k}$ is weakly positive semidefinite:

[TABLE]

for all $n\in{\mathbb{N}}$, $\{x_{j}\}_{j=1}^{n}\in X$, $\{h_{j}\}_{j=1}^{n}\in{\mathcal{H}}$ and $\{t_{j}\}_{j=1}^{n}\in{\mathbb{C}}$.

Define an action of $\Gamma$ on $(X\times{\mathcal{H}})$ in the following way: $\xi\cdot(x,h)=(\xi\cdot x,h)$ for all $\xi\in\Gamma$, $x\in X$ and $h\in{\mathcal{H}}$. Using the $\Gamma$ invariance of $\mathbf{l}$ it follows that $\mathbf{k}$ is $\Gamma$ invariant: letting $\xi\in\Gamma$, $x,y\in X$ and $g,h\in{\mathcal{H}}$ we have

[TABLE]

By Theorem 4.3, there exists a minimal weak VE-space linearisation $({\mathcal{E}};\pi;V)$ of $\mathbf{k}$ and the action of $\Gamma$ on $(X\times{\mathcal{H}})$. By construction, see (3.14), it is clear that $V(x,h)$ depends linearly on $h\in{\mathcal{H}}$, therefore, for each $x\in X$ a linear operator of VE-spaces $\tilde{V}(x)\colon{\mathcal{H}}\rightarrow{\mathcal{E}}$ can be defined by $\tilde{V}(x)h=V(x,h)$.

We now have $[\tilde{V}(x)h,\tilde{V}(y)g]_{{\mathcal{E}}}=\mathbf{k}((x,h),(y,g))=[\mathbf{l}(y,x)h,g]_{{\mathcal{H}}}$ for all $x,y\in X$ and $h,g\in{\mathcal{H}}$. By the minimality of ${\mathcal{E}}$, it follows that $\tilde{V}(x)$ is an adjointable operator with $\tilde{V}(y)^{*}\tilde{V}(x)=\mathbf{l}(y,x)$ for all $x,y\in X$.

On the other hand, we have $\pi(\xi)V(x,h)=V(\xi\cdot x,h)=\tilde{V}(\xi\cdot x)h$ for all $h\in{\mathcal{H}}$ and hence $\pi(\xi)\tilde{V}(x)=\tilde{V}(\xi\cdot x)$ for all $\xi\in\Gamma$ and $x\in X$, showing that $({\mathcal{E}};\pi;\tilde{V})$ is a minimal invariant ${\mathcal{L}}^{*}({\mathcal{H}})$-valued VE-space linearisation of the kernel $\mathbf{l}$ and the action of $\Gamma$ on $X$.

(2)$\Rightarrow$(3). Let $(\tilde{{\mathcal{E}}};\tilde{\pi};\tilde{V})$ be an invariant ${\mathcal{L}}^{*}({\mathcal{H}})$-valued VE-space linearisation of the kernel $\mathbf{l}$, hence $\mathbf{l}(x,y)=\tilde{V}(x)^{*}\tilde{V}(y)$ for all $x,y\in X$. Define $V\colon(X\times{\mathcal{H}})\rightarrow\tilde{{\mathcal{E}}}$ by

[TABLE]

We also have

[TABLE]

hence $\tilde{\pi}(\xi)$ leaves $\tilde{{\mathcal{E}}}_{0}=\operatorname{Lin}V(X,{\mathcal{H}})$ invariant for all $\xi\in\Gamma$. In the following, we denote by the same symbol $\tilde{\pi}\colon\Gamma\rightarrow{\mathcal{L}}^{*}(\tilde{{\mathcal{E}}}_{0})$, the $*$-representation viewed as $\tilde{\pi}(\gamma)\colon\tilde{{\mathcal{E}}}_{0}\rightarrow\tilde{{\mathcal{E}}}_{0}$ for all $\gamma\in\Gamma$. Then $(\tilde{{\mathcal{E}}}_{0};\tilde{\pi};V)$ is a minimal invariant weak VE-space linearisation for the kernel $\mathbf{k}\colon(X\times{\mathcal{H}})\times(X\times{\mathcal{H}})\rightarrow Z$ defined by

[TABLE]

and the action of $\Gamma$ on $(X\times{\mathcal{H}})$ given by

[TABLE]

By Theorem 4.3, there exists a minimal weak $Z$-reproducing kernel VE-space ${\mathcal{R}}\subseteq Z^{X\times{\mathcal{H}}}$, with reproducing kernel $\mathbf{k}$, and a $*$-representation $\rho\colon\Gamma\rightarrow{\mathcal{L}}^{*}({\mathcal{R}})$ such that $\rho(\xi)\mathbf{k}_{(x,h)}=\mathbf{k}_{\xi\cdot(x,h)}$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$. As the proof of Theorem 4.3 shows, without loss of generality we can assume that ${\mathcal{R}}$ is the collection of all maps $X\times{\mathcal{H}}\rightarrow Z$ defined by $X\times{\mathcal{H}}\ni(x,h)\mapsto[\tilde{V}(x)h,f]_{\tilde{{\mathcal{E}}}}$, where $f\in\tilde{{\mathcal{E}}}_{0}$, which provides an identification of ${\mathcal{R}}$ with $\tilde{{\mathcal{E}}}_{0}$ by the formula

[TABLE]

Consequently, for each $f\in{\mathcal{R}}$ and $x\in X$, there exists a unique vector $\tilde{f}(x)=\tilde{V}(x)^{*}f\in{\mathcal{H}}$ such that

[TABLE]

which gives rise to a map ${\mathcal{R}}\ni f\mapsto\tilde{f}\in{\mathcal{H}}^{X}$. Let $\tilde{{\mathcal{R}}}$ be the vector space of all $\tilde{f}$, for $f\in{\mathcal{R}}$. Since, by the reproducing property of the kernel $\mathbf{k}$ and (5.6) we have

[TABLE]

taking into account the reproducing property of the kernel $\mathbf{l}$, it follows that $\mathbf{l}_{x}h=\widetilde{\mathbf{k}_{(x,h)}}\in\tilde{{\mathcal{R}}}$ for all $x\in X$ and $h\in{\mathcal{H}}$.

It is easy to see that the map $U\colon{\mathcal{R}}\ni f\rightarrow\tilde{f}\in\tilde{{\mathcal{R}}}$ is linear, one-to-one, and onto. Therefore, defining $[\tilde{f},\tilde{g}]_{\tilde{{\mathcal{R}}}}:=[f,g]_{{\mathcal{R}}}$ makes $\tilde{{\mathcal{R}}}$ a VE-space, and $U$ becomes a unitary operator of VE-spaces. Defining $\tilde{\rho}:=U\rho U^{*}$, the pair $(\tilde{{\mathcal{R}}},\tilde{\rho})$ has all the required properties.

(3)$\Rightarrow$(1). Assume that $(\tilde{{\mathcal{R}}};\tilde{\rho})$ is a pair consisting of an ${\mathcal{L}}^{*}({\mathcal{H}})$-reproducing kernel VE-space of $\mathbf{l}$ and a $*$-representation $\tilde{\rho}:\Gamma\rightarrow{\mathcal{L}}^{*}(\tilde{{\mathcal{R}}})$ such that $\rho(\xi)\mathbf{l}_{x}h=\mathbf{l}_{\xi\cdot x}h$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$. We have

[TABLE]

for all $n\in{\mathbb{N}}$, $\{x_{i}\}_{i=1}^{n}\in X$, $\{h_{i}\}_{i=1}^{n}\in{\mathcal{H}}$. Therefore $\mathbf{l}$ is positive semidefinite in the sense of (5.1). Moreover, by (hrk3)

[TABLE]

for all $x,y\in X$ and $g,h\in{\mathcal{H}}$, and the invariance of the kernel $\mathbf{l}$ is proven. ∎

Remark 5.2**.**

The crucial point in the proof of the implication (2)$\Rightarrow$(3) of Theorem 5.1 is the proof of (5.6) which we obtained as a consequence of the identification of ${\mathcal{R}}$ with $\tilde{\mathcal{E}}_{0}$. In the following we show that there is a direct proof of (5.6), without using this identification.

By minimality, ${\mathcal{R}}=\operatorname{Lin}\{\mathbf{k}_{(x,h)}\mid x\in X,\ h\in{\mathcal{H}}\}$ so let $f=\sum_{j=1}^{n}\alpha_{j}\mathbf{k}_{(y_{j},g_{j})}$ for some $n\in{\mathbb{N}}$, $y_{1},\ldots,y_{n}\in X$, $g_{1},\ldots,g_{n}\in{\mathcal{H}}$ and $\alpha_{1},\ldots,\alpha_{n}\in{\mathbb{C}}$, be an arbitrary element $f\in{\mathcal{R}}$. Then, for any $x\in X$ and $h\in{\mathcal{H}}$, we have

[TABLE]

hence, letting $\tilde{f}(x)=V(x)^{*}\sum_{j=1}^{n}\alpha_{j}\tilde{V}(y_{j})g_{j}$, (5.6) holds.

5.2. Invariant Kernels with Values Continuously Adjointable Operators.

In this subsection we show that Theorem 2.10 in [4] can be recovered as a special case of Theorem 4.6. We first review definitions in [4] that we will use in this subsection.

Let $X$ be a nonempty set and let ${\mathcal{H}}$ be a VH-space over an admissible space $Z$. In this subsection we will consider kernels $\mathbf{k}\colon X\times X\rightarrow{\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$. Such a kernel $k$ is called positive semidefinite if it is $n$-positive for all natural numbers $n$, in the sense of (5.1).

A ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-valued VH-space linearisation of $\mathbf{k}$, or ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-valued VH-space Kolmogorov decomposition of $\mathbf{k}$, is a pair $({\mathcal{K}};V)$, subject to the following conditions:

• (vhl1)

${\mathcal{K}}$ is a VH-space over $Z$.

• (vhl2)

$V\colon X\rightarrow{\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}},{\mathcal{K}})$ satisfies $\mathbf{k}(x,y)=V(x)^{*}V(y)$ for all $x,y\in X$.

$({\mathcal{K}};V)$ is called topologically minimal if

• (vhl3)

$\operatorname{Lin}V(X){\mathcal{H}}$ is dense in ${\mathcal{K}}$.

We call $\mathbf{k}$ $\Gamma$-invariant if

[TABLE]

A triple $({\mathcal{K}};\pi;V)$ is called a $\Gamma$-invariant ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-valued VH-space linearisation for $\mathbf{k}$ if

• (ihl1)

$({\mathcal{K}};V)$ is an ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-valued VH-space linearisation of $\mathbf{k}$.

• (ihl2)

$\pi\colon\Gamma\rightarrow{\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{K}})$ is a $*$-representation.

• (ihl3)

$V(\xi\cdot x)=\pi(\xi)V(x)$ for all $\xi\in\Gamma$ and all $x\in X$.

Also, $({\mathcal{K}};\pi;V)$ is topologically minimal if the ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-VH-space linearisation $({\mathcal{K}};V)$ is topologically minimal, that is, ${\mathcal{K}}$ is the closure of the linear span of $V(X){\mathcal{H}}$.

A VH-space ${\mathcal{R}}$ over the ordered $*$-space $Z$ is called a ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-reproducing kernel VH-space on $X$ if there exists a Hermitian kernel $\mathbf{k}\colon X\times X\rightarrow{\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$ such that the following axioms are satisfied:

• (rkh1)

${\mathcal{R}}$ is a subspace of ${\mathcal{H}}^{X}$, with all algebraic operations.

• (rkh2)

For all $x\in X$ and all $h\in{\mathcal{H}}$, the ${\mathcal{H}}$-valued function $\mathbf{k}_{x}h=\mathbf{k}(\cdot,x)h\in{\mathcal{R}}$.

• (rkh3)

For all $f\in{\mathcal{R}}$ we have $[f(x),h]_{\mathcal{H}}=[f,\mathbf{k}_{x}h]_{\mathcal{R}}$, for all $x\in X$ and $h\in{\mathcal{H}}$.

• (rkh4)

For all $x\in X$ the evaluation operator ${\mathcal{R}}\ni f\mapsto f(x)\in{\mathcal{H}}$ is continuous.

In this operator valued setting, let us note the appearance of the axiom (rkh4) which makes a difference with classical cases, see [4] for some results pointing out its significance.

Theorem 5.3** (Theorem 2.10 in [4]).**

Let $\Gamma$ be a $*$-semigroup that acts on the nonempty set $X$ and let $\mathbf{l}\colon X\times X\rightarrow{\mathcal{L}}_{\mathrm{c}}^{*}({\mathcal{H}})$ be a kernel, for some VH-space ${\mathcal{H}}$ over an admissible space $Z$. Then the following assertions are equivalent:

• (1)

$\mathbf{l}$* has the following properties:*

• (a)

$\mathbf{l}$* is positive semidefinite, in the sense of (5.1), and invariant under the action of $\Gamma$ on $X$, that is, (4.2) holds.*

• (b)

For any $\xi\in\Gamma$ and any seminorm $p\in S(Z)$, there exists a seminorm $q\in S(Z)$ and a constant $c_{p}(\xi)\geq 0$ such that for all $n\in{\mathbb{N}}$, $\{h_{i}\}_{i=1}^{n}\in{\mathcal{H}}$, $\{x_{i}\}_{i=1}^{n}\in X$ we have

[TABLE]

• (c)

For any $x\in X$ and any seminorm $p\in S(Z)$, there exists a seminorm $q\in S(Z)$ and a constant $c_{p}(x)\geq 0$ such that for all $n\in{\mathbb{N}}$, $\{y_{i}\}_{i=1}^{n}\in X$, $\{h_{i}\}_{i=1}^{n}\in{\mathcal{H}}$ we have

[TABLE]

• (2)

$\mathbf{l}$* has a $\Gamma$-invariant ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-valued VH-space linearisation $({\mathcal{K}};\pi;V)$.*

• (3)

$\mathbf{l}$* admits an ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-reproducing kernel VH-space ${\mathcal{R}}$ and there exists a $*$-representation $\rho\colon\Gamma\rightarrow{\mathcal{L}}_{\mathrm{c}}^{*}({\mathcal{R}})$ such that $\rho(\xi)\mathbf{l}_{x}h=\mathbf{l}_{\xi\cdot x}h$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$.*

In addition, in case any of the assertions (1), (2), or (3) holds, then a minimal  $\Gamma$-invariant ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-valued VH-space linearisation of $\mathbf{l}$ can be constructed, and the pair $({\mathcal{R}};\rho)$ as in assertion (3) can be chosen with ${\mathcal{R}}$ topologically minimal as well.

Proof.

(1)$\Rightarrow$(2). Define the kernel $\mathbf{k}\colon(X\times{\mathcal{H}})\times(X\times{\mathcal{H}})\rightarrow Z$ by

[TABLE]

As in the proof of the implication (1)$\Rightarrow$(2) of Theorem 5.4, $\mathbf{k}$ is weakly positive semidefinite and invariant under the action of $\Gamma$ on $X\times{\mathcal{H}}$ given by $\xi\cdot(x,h)=(\xi\cdot x,h)$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$ In order to see that this kernel satisfies the property 1.(c) of Theorem 4.5, observe that for all $n\in{\mathbb{N}}$, $\{t_{i}\}_{i=1}^{n}\subset{\mathbb{C}}$, $\alpha\in\Gamma$, and $p\in S(Z)$, by assumption, see property 1.(b), there exists $q\in S(Z)$ and $c(\alpha)\geq 0$, we have

[TABLE]

By Theorem 4.5, there exists a minimal weak VH-space linearisation $({\mathcal{K}};\pi;V)$ of $\mathbf{k}$ and the action of $\Gamma$ on $(X\times{\mathcal{H}})$. Same arguments as in the proof of the implication (1)$\Rightarrow$(2) of Theorem 5.4 show that, for any $x\in X$, there exists an adjointable operator of VE-spaces $\tilde{V}(x)\colon{\mathcal{H}}\rightarrow{\mathcal{K}}_{0}$, given by $\tilde{V}(x)h\colon=V(x,h)$ for $x\in X$ and $h\in{\mathcal{H}}$, where ${\mathcal{K}}_{0}\colon=\operatorname{Lin}V(X){\mathcal{H}}$, with the property that $\tilde{V}(x)^{*}\tilde{V}(y)=\mathbf{l}(x,y)$ for all $x,y\in X$. Arguing as in the proof of the implication (1)$\Rightarrow$(2) of Theorem 2.10 of [4], it follows that $\tilde{V}(x)\in{\mathcal{L}}_{\mathrm{c}}^{*}({\mathcal{H}},{\mathcal{K}}_{0})$. Now using the boundedness condition (c), for any $p\in S(Z)$ there exist $q\in S(Z)$ and $c_{p}(x)\geq 0$ such that, for all $\sum_{i=1}^{n}V(y_{i})h_{i}\in{\mathcal{K}}_{0}$ we have

[TABLE]

hence $\tilde{V}(x)^{*}\in{\mathcal{L}}_{\mathrm{c}}^{*}({\mathcal{K}}_{0},{\mathcal{H}})$ for any $x\in X$. Consequently, $\tilde{V}(x)^{*}$ extends uniquely to an operator $\tilde{V}(x)^{*}\in{\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{K}},{\mathcal{H}})$ for each $x\in X$. It follows that $({\mathcal{K}};\pi;\tilde{V})$ is an invariant ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-valued VH-space linearisation of the kernel $\mathbf{l}$ and the action of $\Gamma$ on $X$.

(2)$\Rightarrow$(3). Let $(\tilde{{\mathcal{K}}};\tilde{\pi};\tilde{V})$ be an invariant ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-valued VH-space linearisation of the kernel $\mathbf{l}$. In order to avoid repetition, we use some facts obtained during the proof of the implication (2)$\Rightarrow$(3) of Theorem 5.1. Define $V\colon(X\times{\mathcal{H}})\rightarrow\tilde{{\mathcal{K}}}$ by $V(x,h)=\tilde{V}(x)h$ for all $x\in X$ and $h\in{\mathcal{H}}$. Letting $\tilde{{\mathcal{K}}}_{0}=\overline{\operatorname{Lin}V(X,{\mathcal{H}})}\subseteq\tilde{{\mathcal{K}}}$, similarly we see that $(\overline{\operatorname{Lin}V(X,{\mathcal{H}})};\tilde{\pi}_{0};V)$ is a topological minimal invariant weak VH-space linearisation for the kernel $\mathbf{k}\colon(X\times{\mathcal{H}})\times(X\times{\mathcal{H}})\rightarrow Z$ defined by $\mathbf{k}((x,h),(y,g))=[V(x,h),V(y,g)]$ for all $x,y\in X$ and $h,g\in{\mathcal{H}}$ and the action of $\Gamma$ on $(X\times{\mathcal{H}})$ defined by $\xi\cdot(x,h)=(\xi\cdot x,h)$ for all $\xi\in\Gamma$, $x\in X$ and $h\in{\mathcal{H}}$.

By Theorem 4.5 there exists a topologically minimal weak $Z$-reproducing kernel VH-space ${\mathcal{R}}$ and a $*$-representation $\rho\colon\Gamma\rightarrow{\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{R}})$ such that $\rho(\xi)\mathbf{k}_{(x,h)}=\mathbf{k}_{\xi\cdot(x,h)}$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$. The rest of the proof is similar with the end of the proof of the implication (2)$\Rightarrow$(3) as in Theorem 5.1. We show that, for each $f\in{\mathcal{R}}$ and $x\in X$ there exists a unique element $\tilde{f}(x)$ such that (5.6) holds and, consequently, this gives rise to a map ${\mathcal{R}}\ni f\mapsto Uf=\tilde{f}\in{\mathcal{H}}^{X}$, which is linear and bijective between ${\mathcal{R}}$ and its range $\tilde{{\mathcal{R}}}\subseteq{\mathcal{H}}^{X}$. Letting $[\tilde{f},\tilde{g}]_{\tilde{{\mathcal{R}}}}=[f,g]_{\mathcal{R}}$ for all $f,g\in{\mathcal{R}}$, $\tilde{{\mathcal{R}}}$ becomes an ${\mathcal{H}}$-valued reproducing kernel VH-space with kernel $\mathbf{l}$, and then letting $\tilde{\rho}\colon=U\rho U^{*}$, $(\tilde{{\mathcal{R}}},\tilde{\rho})$ is a pair having all the required properties.

(3)$\Rightarrow$(1). Assume that the pair $(\tilde{{\mathcal{R}}};\tilde{\rho})$ consists of an ${\mathcal{L}}^{*}_{\mathrm{c}}({\mathcal{H}})$-valued reproducing kernel VH-space of $\mathbf{l}$ and a $*$-representation $\tilde{\rho}$ of $\Gamma$ on ${\mathcal{L}}_{\mathrm{c}}^{*}(\tilde{{\mathcal{R}}})$ such that $\rho(\xi)\mathbf{l}_{x}h=\mathbf{l}_{\xi\cdot x}h$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$. Similarly as in the proof of the implication (3)$\Rightarrow$(1) of Theorem 5.1, the kernel $\mathbf{l}$ is shown to be positive semidefinite and invariant under the action of $\Gamma$ on $X$. On the other hand, the inequalities (b) and (c) are obtained from the continuity of the operator $\rho(\xi)\colon{\mathcal{R}}\rightarrow{\mathcal{R}}$, for any $\xi\in\Gamma$ and, respectively, from the continuity of the evaluation operator $E_{x}\colon{\mathcal{R}}\rightarrow{\mathcal{H}}$, for any $x\in X$. ∎

5.3. Invariant Kernels with Values Boundedly Adjointable Operators.

We show that Theorem 4.2 in [11] is a special case of Theorem 4.5. We review necessary definitions in [11].

Given a ${\mathcal{B}}^{*}({\mathcal{H}})$-valued kernel $\mathbf{l}$ on a nonempty set $X$, where ${\mathcal{H}}$ is a VH-space over the admissible space $Z$, a ${\mathcal{B}}^{*}({\mathcal{H}})$-valued VH-space linearisation of $\mathbf{l}$ is a pair $(\tilde{{\mathcal{K}}};\tilde{V})$ with

• (hvhl1)

$\tilde{{\mathcal{K}}}$ is a VH-space over $Z$.

• (hvhl2)

$\tilde{V}\colon X\rightarrow{\mathcal{B}}^{*}({\mathcal{H}};{\mathcal{K}})$ satisfies $\mathbf{l}(x,y)=\tilde{V}(x)^{*}\tilde{V}(y)$ for all $x,y\in X$.

If $\Gamma$ is a $*$-semigroup acting on $X$, $(\tilde{{\mathcal{K}}};\tilde{\pi};\tilde{V})$ is called an invariant ${\mathcal{B}}^{*}({\mathcal{H}})$-valued VH-space linearisation of the kernel $\mathbf{l}$ and the action of $\Gamma$ on $X$, if, in addition to (hvhl1) and (hvhl2), we have,

• (hvhl3)

$\tilde{\pi}\colon\Gamma\rightarrow{\mathcal{B}}^{*}(\tilde{{\mathcal{K}}})$ is a $*$-representation.

• (hvhl4)

$\tilde{V}(\xi\cdot x)=\tilde{\pi}(\xi)V(x)$ for every $\xi\in\Gamma$, $x\in X$.

If we have

• (hvhl5)

$\mathrm{Lin}\tilde{V}(X){\mathcal{H}}$ is dense in $\tilde{{\mathcal{K}}}$,

then $(\tilde{{\mathcal{K}}};\tilde{\pi};\tilde{V})$ is called topologically minimal.

Given a nonempty set $X$ and a VH-space ${\mathcal{H}}$ over the admissible space $Z$, a VH-space $\tilde{{\mathcal{R}}}$ over $Z$ is called a ${\mathcal{B}}^{*}({\mathcal{H}})$-valued reproducing kernel VH-space on $X$ if there exists a kernel $\mathbf{l}\colon X\times X\rightarrow{\mathcal{B}}^{*}({\mathcal{H}})$ such that

• (hrk1)

$\tilde{{\mathcal{R}}}$ is a subspace of ${\mathcal{H}}^{X}$ with all algebraic operations.

• (hrk2)

$\mathbf{l}_{x}h=\mathbf{l}(\cdot,x)h\in\tilde{{\mathcal{R}}}$ for all $x\in X$, $h\in{\mathcal{H}}$.

• (hrk3)

$[f(x),h]_{{\mathcal{H}}}=[f,\mathbf{l}_{x}h]_{\tilde{{\mathcal{R}}}}$ holds for all $f\in\tilde{{\mathcal{R}}}$, $x\in X$ and $h\in{\mathcal{H}}$.

$\tilde{\mathcal{R}}$ is called topologically minimal if

• (hrk4)

$\mathrm{Lin}\{\mathbf{l}_{x}h\mid x\in X,h\in{\mathcal{H}}\}$ is dense in $\tilde{{\mathcal{R}}}$,

and, in this case, $\tilde{\mathcal{R}}$ is uniquely determined by the kernel $\mathbf{l}$.

Theorem 5.4** (Theorem 4.2 in [11]).**

Let $\Gamma$ be a $*$-semigroup acting on a nonempty set $X$, ${\mathcal{H}}$ be a VH-space on an admissible space $Z$, and $\mathbf{l}:X\times X\rightarrow{\mathcal{B}}^{*}({\mathcal{H}})$ be a kernel. Then the following are equivalent:

• (1)

$\mathbf{l}$* has the following properties:*

• (a)

$\mathbf{l}$* is positive semidefinite.*

• (b)

$\mathbf{l}$* is invariant under the action of $\Gamma$ on $X$.*

• (c)

For any $\alpha\in\Gamma$ there exists $c(\alpha)\geq 0$ such that, for all $n\in{\mathbb{N}}$, $x_{1},x_{2},\cdots x_{n}\in X$, $h_{1},h_{2},\cdots,h_{n}\in{\mathcal{H}}$, we have

[TABLE]

• (2)

$\mathbf{l}$* has an invariant ${\mathcal{B}}^{*}({\mathcal{H}})$-valued VH-space linearisation $(\tilde{{\mathcal{E}}};\tilde{\pi};\tilde{V})$.*

• (3)

$\mathbf{l}$* admits a ${\mathcal{B}}^{*}({\mathcal{H}})$-reproducing kernel VH-space $\tilde{{\mathcal{R}}}$ and there exists a $*$-representation $\tilde{\rho}:\Gamma\rightarrow{\mathcal{B}}^{*}(\tilde{{\mathcal{R}}})$ such that $\tilde{\rho}(\xi)\mathbf{l}_{x}h=\mathbf{l}_{\xi\cdot x}h$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$.*

Moreover, in case any of the assertions (1), (2) or (3) holds, a topologically minimal invariant ${\mathcal{B}}^{*}({\mathcal{H}})$-valued VH-space linearisation can be constructed.

Proof.

(1)$\Rightarrow$(2). Define the kernel $\mathbf{k}\colon(X\times{\mathcal{H}})\times(X\times{\mathcal{H}})\rightarrow Z$ by

[TABLE]

for all $x,y\in X$ and $h,g\in{\mathcal{H}}$. Then $\mathbf{k}$ is weakly positive semidefinite and invariant under the action of $\Gamma$ on $X\times{\mathcal{H}}$ given by $\xi\cdot(x,h)=(\xi\cdot x,h)$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$, as in the proof of (1)$\Rightarrow$(2) of Theorem 5.1. To see that this kernel satisfies condition 1.(c) of Theorem 4.5, we use the assumption (c) and get that, for any $\alpha\in\Gamma$ there exists $c(\alpha)\geq 0$ such that, for all $n\in{\mathbb{N}}$, $\{t_{i}\}_{i=1}^{n}\subset{\mathbb{C}}$, we have

[TABLE]

By Theorem 4.5, there exists a minimal weak VH-space linearisation $({\mathcal{K}};V;\pi)$ of $\mathbf{k}$ and the action of $\Gamma$ on $(X\times{\mathcal{H}})$. Same arguments as in proof of (1)$\Rightarrow$(2) of Theorem 5.1 gives an adjointable operator of VE-spaces $\tilde{V}(x)\colon{\mathcal{H}}\rightarrow{\mathcal{K}}_{0}$, given by $\tilde{V}(x)h\colon=V(x,h)$ for $x\in X$ and $h\in{\mathcal{H}}$, where ${\mathcal{K}}_{0}\colon=\mathrm{Lin}V(X){\mathcal{H}}$, with the property that $\tilde{V}(x)^{*}\tilde{V}(y)=\mathbf{l}(x,y)$ for all $x,y\in X$. Arguing as in the proof of Theorem 3.3 of [11], it follows that $\tilde{V}(x)\in{\mathcal{B}}^{*}({\mathcal{H}},{\mathcal{K}}_{0})$ and $\tilde{V}(x)^{*}\in{\mathcal{B}}^{*}({\mathcal{K}}_{0},{\mathcal{H}})$. Hence $\tilde{V}(x)^{*}$ extends uniquely to an operator $V(x)^{*}\in{\mathcal{B}}^{*}({\mathcal{K}},{\mathcal{H}})$ for each $x\in X$. It follows that $({\mathcal{K}};\pi;\tilde{V})$ is a topologically minimal invariant ${\mathcal{B}}^{*}({\mathcal{H}})$-valued VH-space linearisation of the kernel $\mathbf{l}$ and the action of $\Gamma$ on $X$.

(2)$\Rightarrow$(3). Let $(\tilde{{\mathcal{K}}};\tilde{\pi};\tilde{V})$ be an invariant ${\mathcal{B}}^{*}({\mathcal{H}})$ VH-space linearisation of the kernel $\mathbf{l}$. We essentially use Theorem 4.5 with details very close to the proof of the implication (2)$\Rightarrow$(3) of Theorem 5.3, with the difference that we obtain bounded adjointable operators instead of continuously adjointable operators. Define $V\colon(X\times{\mathcal{H}})\rightarrow\tilde{{\mathcal{K}}}$ by $V(x,h)=\tilde{V}(x)h$ for all $x\in X$ and $h\in{\mathcal{H}}$. We also have $\tilde{\pi}(\xi)\tilde{V}(x)h=\tilde{V}(\xi\cdot x)h=V(\xi\cdot x,h)$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$. Then $(\overline{\operatorname{Lin}(V(X,{\mathcal{H}})};V;\tilde{\pi})$ is a topologically minimal weak invariant VH-space linearisation for the kernel $\mathbf{k}\colon(X\times{\mathcal{H}})\times(X\times{\mathcal{H}})\rightarrow Z$ defined by $\mathbf{k}((x,h),(y,g))=[V(x,h),V(y,g)]$, for all $x,y\in X$ and $h,g\in{\mathcal{H}}$, and the action of $\Gamma$ on $(X\times{\mathcal{H}})$ defined by $\xi\cdot(x,h)=(\xi\cdot x,h)$, for all $\xi\in\Gamma$, $x\in X$, and $h\in{\mathcal{H}}$.

By Theorem 4.5 there exists a weak $Z$-reproducing kernel VH-space ${\mathcal{R}}$ and a $*$-representation $\rho\colon\Gamma\rightarrow{\mathcal{B}}^{*}({\mathcal{R}})$ such that $\rho(\xi)\mathbf{k}_{(x,h)}=\mathbf{k}_{\xi\cdot(x,h)}$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$. Define $\tilde{f}\colon X\rightarrow{\mathcal{H}}$ as follows: for each $x\in X$ let $\tilde{f}(x)\in{\mathcal{H}}$ be the unique element satisfying $[\tilde{f}(x),h]_{{\mathcal{H}}}=f(x,h)$ for all $h\in{\mathcal{H}}$ and let $\tilde{{\mathcal{R}}}$ be the vector space of all $\tilde{f}$, when $f\in{\mathcal{R}}$. Since we have

[TABLE]

it follows that $\mathbf{l}_{x}h=\widetilde{\mathbf{k}_{(x,h)}}\in\tilde{{\mathcal{R}}}$ for all $x\in X$ and $h\in{\mathcal{H}}$.

It is easy to check that the map $U\colon{\mathcal{R}}\ni f\rightarrow\tilde{f}\in\tilde{{\mathcal{R}}}$ is linear, one-to-one, and onto. Therefore, defining $[\tilde{f},\tilde{g}]_{\tilde{{\mathcal{R}}}}:=[f,g]_{{\mathcal{R}}}$ makes $\tilde{{\mathcal{R}}}$ a ${\mathcal{B}}^{*}({\mathcal{H}})$-reproducing kernel VH-space with reproducing kernel $\mathbf{l}$, and $U$ becomes a unitary operator of VH-spaces. Defining $\tilde{\rho}:=U\rho U^{*}$, the pair $(\tilde{{\mathcal{R}}},\tilde{\rho})$ has all the required properties.

(3)$\Rightarrow$(1). Assume that $(\tilde{{\mathcal{R}}};\tilde{\rho})$ is a ${\mathcal{B}}^{*}({\mathcal{H}})$-reproducing kernel VH-space of $\mathbf{l}$ with a representation $\tilde{\rho}$ of $\Gamma$ on ${\mathcal{B}}^{*}(\tilde{{\mathcal{R}}})$ such that $\rho(\xi)\mathbf{l}_{x}h=\mathbf{l}_{\xi\cdot x}h$ for all $\xi\in\Gamma$, $x\in X$, $h\in{\mathcal{H}}$. Similarly as in proof of the implication (3)$\Rightarrow$(1) of Theorem 5.1, the kernel $\mathbf{l}$ is shown to be positive semidefinite and invariant under the action of $\Gamma$ on $X$. On the other hand, using the fact that the linear operator $\tilde{\rho}(\xi)\colon{\mathcal{H}}\rightarrow{\mathcal{H}}$ is bounded for all $\xi\in\Gamma$, it follows that, for any $\xi\in\Gamma$, there exists $c(\xi)\geq 0$ such that, for all $\xi\in\Gamma$, $n\in{\mathbb{N}}$, $\{x_{i}\}_{i=1}^{n}\in X$ and $\{h_{i}\}_{i=1}^{n}\in{\mathcal{H}}$, we have

[TABLE]

hence $\mathbf{l}$ has the property (c). ∎

5.4. Positive Semidefinite ${\mathcal{L}}({\mathcal{X}},{\mathcal{X}}_{Z}^{\prime})$ Valued Maps on $*$-Semigroups.

In this subsection we obtain stronger versions of Theorem 3.1 and Theorem 4.2 in [27] as applications of Theorem 4.3 and, respectively, of Theorem 4.5. We first reorganise some definitions from [27] and [13].

Let ${\mathcal{X}}$ be a vector space, and $Z$ be an ordered $*$-space. By ${\mathcal{X}}_{Z}^{\prime}$ we denote the space of all conjugate linear functions from ${\mathcal{X}}$ to $Z$ and call it the algebraic conjugate $Z$-dual space. Let ${\mathcal{L}}({\mathcal{X}},{\mathcal{X}}_{Z}^{\prime})$ denote the vector space of all linear operators $T\colon{\mathcal{X}}\rightarrow{\mathcal{X}}_{Z}^{\prime}$. For any VE-space ${\mathcal{E}}$ over $Z$ and any linear operator $A\colon{\mathcal{X}}\rightarrow{\mathcal{E}}$, we define a linear operator $A^{\prime}\colon{\mathcal{E}}\rightarrow{\mathcal{X}}_{Z}^{\prime}$, called the algebraic $Z$-adjoint operator, by

[TABLE]

If $\Gamma$ is a $*$-semigroup, a map $T\colon\Gamma\rightarrow{\mathcal{L}}({\mathcal{X}},{\mathcal{X}}_{Z}^{\prime})$ is called ${\mathcal{L}}({\mathcal{X}},{\mathcal{X}}_{Z}^{\prime})$-valued $n$-positive if

[TABLE]

If $T$ is $n$-positive for all $n\in{\mathbb{N}}$ then it is called ${\mathcal{L}}({\mathcal{X}},{\mathcal{X}}_{Z}^{\prime})$-valued positive semidefinite.

Remarks 5.5**.**

With notation as before, let $T\colon\Gamma\rightarrow{\mathcal{L}}({\mathcal{X}},{\mathcal{X}}_{Z}^{\prime})$.

(1) We define a kernel $\mathbf{k}\colon(\Gamma\times{\mathcal{X}})\times(\Gamma\times{\mathcal{X}})\rightarrow Z$ by

[TABLE]

Then for all $n\in{\mathbb{N}}$, all $\alpha_{1},\alpha_{2},\dots,\alpha_{n}\in{\mathbb{C}}$, and all $(s_{i},x_{i})_{i=1}^{n}\in(S\times X)$ we have

[TABLE]

This shows that, for $n\in{\mathbb{N}}$, the map $T$ is $n$-positive if and only if the kernel $\mathbf{k}$ is weakly $n$-positive. In particular, $T$ is positive semidefinite if and only if the kernel $\mathbf{k}$ is weakly positive semidefinite.

(2) Recall that, see (3.2), the kernel $\mathbf{k}$ is Hermitian if $\mathbf{k}((s,x),(t,y))=\mathbf{k}((t,y),(s,x))^{*}$ for all $s,t\in\Gamma$ and all $x,y\in{\mathcal{X}}$. From (5.11) it follows that $\mathbf{k}$ is Hermitian if and only if

[TABLE]

Consequently, by Lemma 3.1 it follows that, if $T$ is $2$-positive, then (5.12) holds.

In addition, if $\Gamma$ has a unit $e=e^{*}$, then (5.11) is equivalent with

[TABLE]

(3) We define a left action of $\Gamma$ on $(\Gamma\times{\mathcal{X}})$ by

[TABLE]

For all $u\in\Gamma$ and all $(s,x)\in\Gamma\times{\mathcal{X}}$ we have

[TABLE]

hence the kernel $\mathbf{k}$ is invariant under the left action of $\Gamma$ on $\Gamma\times{\mathcal{X}}$ defined as in (5.14).

Theorem 5.6**.**

Let $Z$ be an ordered $*$-space, let ${\mathcal{X}}$ be complex vector space with algebraic conjugate $Z$-dual space ${\mathcal{X}}_{Z}^{\prime}$, and consider $T\colon\Gamma\rightarrow{\mathcal{L}}({\mathcal{X}},{\mathcal{X}}_{Z}^{\prime})$, for some $*$-semigroup $\Gamma$ with unit. The following assertions are equivalent:

• (i)

$T$* is positive semidefinite, in the sense of (5.10).*

• (ii)

There exist a VE-space ${\mathcal{E}}$ over $Z$, a unital $*$-representation $\pi\colon\Gamma\rightarrow{\mathcal{L}}^{*}({\mathcal{E}})$, and an operator $A\in{\mathcal{L}}({\mathcal{X}},{\mathcal{E}})$, such that

[TABLE]

If any of the conditions (i) and (ii) holds, then the VE-space ${\mathcal{E}}$ can be chosen minimal in the sense that it coincides with the linear span of $\pi(\Gamma)A{\mathcal{X}}$ and, in this case, it is unique modulo a unitary equivalence.

Proof.

(i)$\Rightarrow$(ii). We consider the kernel $\mathbf{k}\colon\Gamma\times{\mathcal{X}}\rightarrow Z$ as in (5.11) and the left action of $\Gamma$ on $\Gamma\times{\mathcal{X}}$ as in (5.14). By Remark 5.5.(1) and Remark 5.5.(2), $\mathbf{k}$ is a $Z$-valued weakly positive semidefinite kernel invariant under the action of $\Gamma$ as in (5.14) hence, by Theorem 4.3, there exists a minimal invariant weak VE-space linearisation $({\mathcal{E}},\pi,V)$ of $\mathbf{k}$. Since

[TABLE]

it follows that ${\mathcal{X}}\ni x\mapsto V(s,x)\in{\mathcal{E}}$ is linear, for all $s\in\Gamma$. This shows that, we can define $\tilde{V}\colon\Gamma\rightarrow{\mathcal{L}}({\mathcal{X}},{\mathcal{E}})$ by $\tilde{V}(s)x=V(s,x)$, for all $s\in\Gamma$ and all $x\in{\mathcal{X}}$. Taking into account (5.9) it follows that

[TABLE]

hence, letting $A=\tilde{V}(e)\in{\mathcal{L}}({\mathcal{X}},{\mathcal{E}})$ it follows that, for all $s\in\Gamma$ and all $x,y\in{\mathcal{X}}$ we have

[TABLE]

and hence (5.15) is proven. The minimality and the uniqueness property follow by standard arguments that we omit.

(ii)$\Rightarrow$(i). This follows by a standard argument that we omit. ∎

Theorem 5.6 is stronger than Theorem 3.1 in [27] since, in addition to positive semidefiniteness of $T$ they require the condition (5.13) as well. As we have seen in Remark 5.5.(3), this condition is a consequence of the positive semidefiniteness of $T$. Also, the ordered $*$-space $Z$ need not be admissible, actually, the topology of $Z$ does not play any role.

From now on we assume that $Z$ is a topologically ordered $*$-space and that ${\mathcal{X}}$ is a locally bounded topological vector space, that is, in ${\mathcal{X}}$ there exists a bounded neighbourhood of [math]. By ${\mathcal{X}}_{Z}^{*}$ we denote the subspace of ${\mathcal{X}}_{Z}^{\prime}$ of all continuous conjugate linear functions from ${\mathcal{X}}$ to $Z$ and call it the topological conjugate $Z$-dual space. The space ${\mathcal{X}}_{Z}^{*}$ is considered with the topology of uniform convergence on bounded sets, that is, a net $(f_{i})_{i\in{\mathcal{I}}}\in{\mathcal{X}}_{Z}^{*}$ converges to [math] if for any bounded subset $B\subset{\mathcal{X}}$ the $Z$-valued net $(f_{i}(y))_{i\in{\mathcal{I}}}$ converges to [math] uniformly with respect to $y\in B$, equivalently, for any bounded set $B\subset{\mathcal{X}}$, any $p\in S(Z)$ and any $\epsilon>0$, there exists $i_{0}\in{\mathcal{I}}$ such that $i\geq i_{0}$ implies $p(f_{i}(y))<\epsilon$ for all $y\in B$. Let ${\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{X}}_{Z}^{*})$ be the space of all continuous linear operators from ${\mathcal{X}}$ to ${\mathcal{X}}_{Z}^{*}$.

Let ${\mathcal{E}}$ be a VE-space over $Z$, with topology defined as in Subsection 2.3. Following [27] and [14], for any $A\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{E}})$ the topological $Z$-adjoint operator of $A$ is, by definition, the operator $A^{*}\colon{\mathcal{E}}\rightarrow{\mathcal{X}}_{Z}^{*}$ defined by

[TABLE]

By Lemma 2.2 the definition of $A^{*}$ is correct.

Theorem 5.7**.**

Let $\Gamma$ be a $*$-semigroup with unit $e$ and ${\mathcal{X}}$ be a locally bounded topological vector space with topological conjugate $Z$-dual space ${\mathcal{X}}_{Z}^{*}$ for an admissible space $Z$. Let $T\colon\Gamma\rightarrow{\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{X}}_{Z}^{\prime})$ subject to the following properties:

• (a)

$T$* is an ${\mathcal{L}}({\mathcal{X}},{\mathcal{X}}_{Z}^{\prime})$-valued positive semidefinite map.*

• (b)

For all $u\in\Gamma$, there is a constant $c(u)\geq 0$ such that for all $n\in{\mathbb{N}}$, all $s_{1},\ldots,s_{n}\in\Gamma$, and all $x_{1},\ldots,x_{n}\in{\mathcal{X}}$, we have

[TABLE]

• (c)

$T(e)\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{X}}_{Z}^{*})$.

Then:

• (i)

There exist a VH-space ${\mathcal{K}}$ over $Z$, a $*$-representation $\pi\colon\Gamma\rightarrow{\mathcal{B}}^{*}({\mathcal{K}})$ and an operator $A\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{K}})$, such that $T_{s}=A^{*}\pi(s)A$ for any $s\in\Gamma$.

• (ii)

$T_{s}\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{X}}_{Z}^{*})$* for all $s\in\Gamma$.*

• (iii)

If $(u_{l})_{l\in{\mathcal{L}}}$ is a net in $\Gamma$ with $\sup_{l\in{\mathcal{L}}}c(u_{l})<\infty$ and $(T_{su_{l}t})_{l\in{\mathcal{L}}}$ converges to $T_{sut}$, for some $u\in\Gamma$ and any $s,t\in\Gamma$, in the weak topology of ${\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{X}}_{Z}^{*})$, then $(\pi(u_{l}))_{l\in{\mathcal{L}}}$ converges to $\pi(u)$ in the weak topology of ${\mathcal{B}}^{*}({\mathcal{K}})$.

Proof.

Define the kernel $\mathbf{k}\colon(\Gamma\times{\mathcal{X}})\times(\Gamma\times{\mathcal{X}})\rightarrow Z$ as in (5.11). By Remark 5.5, it follows that $\mathbf{k}$ is a $Z$-valued weakly positive semidefinite kernel. Next, consider the left action of $\Gamma$ on $(\Gamma\times{\mathcal{X}})$ as in (5.14) and by Remark 5.5 it follows that $\mathbf{k}$ is invariant under this action. In order to show that the property 1.(c) of Theorem 4.5 holds, let $u\in\Gamma$, $n\in{\mathbb{N}}$, $(s_{i},x_{i})_{i=1}^{n}\in(\Gamma\times{\mathcal{X}})$. Then, using (5.17) it follows that

[TABLE]

By Theorem 4.5, there exists a topologically minimal invariant weak VH-space linearisation $({\mathcal{K}};\pi;V)$ of the kernel $\mathbf{k}$. Since $[V(s,x),V(t,y)]_{{\mathcal{K}}}=\mathbf{k}((s,x),(t,y))=(T_{s^{*}t}y)(x)$ for all $s,t\in\Gamma$ and $x,y\in{\mathcal{X}}$, we observe that $V(s,x)$ depends linearly on $x\in{\mathcal{X}}$ for each $s\in\Gamma$. As a consequence, letting $\tilde{V}(s)x=V(s,x)$, for all $x\in{\mathcal{X}}$, we obtain a linear operator $\tilde{V}(s):{\mathcal{X}}\rightarrow{\mathcal{K}}$ for each $s\in\Gamma$. To see that $\tilde{V}(s)$ is continuous for each $s\in\Gamma$, let $(x_{l})_{l\in{\mathcal{L}}}$ be a net in ${\mathcal{X}}$ converging to [math]. Since ${\mathcal{X}}$ is locally bounded, there exists $B\subset{\mathcal{X}}$ a bounded neighbourhood of [math] and then there exists $l_{1}\in{\mathcal{L}}$ such that $(x_{l})_{l\geq l_{1}}$ is contained in $B$. Since $T_{e}\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{X}}_{Z}^{*})$, taking into account the topology of ${\mathcal{X}}_{Z}^{*}$, given any $\varepsilon>0$ and any $p\in S(Z)$ we can find $l_{2}\in{\mathcal{L}}$ such that ${\mathcal{L}}\ni l\geq l_{2}$ implies $p((T_{e}x_{l})y)<\varepsilon$ for all $y\in B$. Since ${\mathcal{L}}$ is directed, there exists $l_{0}\in{\mathcal{L}}$ with $l_{0}\geq l_{1}$ and $l_{0}\geq l_{2}$. Then, for any $l\geq l_{0}$, by (5.17) and taking into account how the topology of ${\mathcal{K}}$ is defined, see Subsection 2.3, we have

[TABLE]

hence $\tilde{V}(s)\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{K}})$, for any $s\in\Gamma$. In addition, for each $s\in\Gamma$ the operator $\tilde{V}(s)^{*}\in{\mathcal{L}}({\mathcal{K}},{\mathcal{X}}_{Z}^{*})$ is defined as in (5.16).

Letting $A:=\tilde{V}(e)$ we have

[TABLE]

Therefore $A^{*}\pi(s)A=T_{s}\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{X}}_{Z}^{*})$, for all $s\in\Gamma$.

The rest of the proof, which shows that $\pi(u_{l})_{l\in{\mathcal{L}}}$ converges to $\pi(u)$ in the weak topology of ${\mathcal{B}}^{*}({\mathcal{K}})$, as in the second part of the conclusion, uses standard arguments and is the same with that in [27]. For completeness, we present it here. Let ${\mathcal{K}}_{0}:=\mathrm{Lin}V(\Gamma\times{\mathcal{X}})$. By minimality, ${\mathcal{K}}_{0}$ is dense in ${\mathcal{K}}$. Let $e,f\in{\mathcal{K}}_{0}$, with $e=\sum_{i=1}^{n}\alpha_{i}V(s_{i},x_{i})$ and $f=\sum_{j=1}^{m}\beta_{j}V(r_{j},y_{j})$. We have

[TABLE]

by the assumption that $(T_{su_{l}t})_{l\in{\mathcal{L}}}$ converges to $T_{sut}$ for any $s,t\in\Gamma$. Now let $g,h\in{\mathcal{K}}$ and let $(g_{i})_{i\in{\mathcal{I}}}\in{\mathcal{K}}_{0}$ be a net converging to $g$. For any $p\in S(Z)$, $j\in{\mathcal{I}}$ and $l\in{\mathcal{L}}$ we have

[TABLE]

for some constant $c$, where the second inequality follows by the Schwarz type inequality (2.11) and the third inequality by the fact that $\pi(u),\pi(u_{l})\in{\mathcal{B}}^{*}({\mathcal{K}})$ and the assumption that $\sup_{l\in{\mathcal{L}}}c(u_{l})<\infty$. Now that weak convergence was shown in ${\mathcal{K}}_{0}$, a standard argument finishes the proof. ∎

Remarks 5.8**.**

(1) Theorem 5.7 is stronger than Theorem 4.2 of [27], see also the correction in [28], with respect to two aspects: firstly, since they have the additional assumption that (5.13) holds, which is actually a consequence of positive semidefiniteness, as Remark 5.5.(2) shows, and secondly since their assumption $T_{s}\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{X}}_{Z}^{*})$, for all $s\in\Gamma$, is actually a consequence of the weaker one $T_{e}\in{\mathcal{L}}_{\mathrm{c}}({\mathcal{X}},{\mathcal{X}}_{Z}^{*})$, as the proof of Theorem 5.7 shows.

(2) It is easy to see that, there is a ”converse” to Theorem 5.7 in the sense that, if assertion (i) is assumed, then assertions (a), (b), (c), and (ii) are obtained as consequences.