Weighted Translation Semigroups: Multivariable Case

TL;DR

This paper extends the theory of weighted translation semigroups to multivariable settings, developing analytic models, spectral descriptions, and examples in higher dimensions, generalizing previous one-variable work.

Contribution

It introduces a multivariable generalization of weighted translation semigroups, constructs an analytic model, and explores spectral properties in higher-dimensional spaces.

Findings

Developed a toral analogue of the analytic model.

Described the spectral picture of multivariable semigroups.

Provided numerous examples in multi-variable cases.

Abstract

M. Embry and A. Lambert initiated the study of a weighted translation semigroup $\{S_t\}$ in ${\cal B}(L^2({\mathbb R_+})),$ with a view to explore a continuous analogue of a weighted shift operator. We continued the work, characterized some special types of semigroups and developed an analytic model for the left invertible weighted translation semigroup. The present paper deals with the generalization of the weighted translation semigroup in multi-variable set up. We develop the toral analogue of the analytic model and also describe the spectral picture. We provide many examples of weighted translation semigroups in multi-variable case. Further, we replace the space $L^2({\mathbb R_+})$ by $L^2({\mathbb R_+^d})$ and explore the properties of weighted translation semigroup $\{S_{\bar{t}}\}$ in ${\cal B}(L^2({\mathbb R_+^d})),$ in both one and multi variable cases.

Equations

$$[H_j^{\ast },H_i]=H_j^{\ast }{H}_i-H_i{H}_j^{\ast }.$$

$$Q_s(X)=\sum _{i=1}^d{T}_i^{\ast }X{T}_i(X\in \mathcal{B}(H)).$$

$$B_p(Q_s)=\sum _{q\in \mathbb{N},0\le q\le p}(-1{)}^q(\genfrac{}{}{0ex}{}{p}{q})Q_s^q(I)\le 0(resp.\ge 0),$$

$$Q_t=(Q_1,\cdots ,Q_d),Q_i(X)=T_i^{\ast }X{T}_i(X\in \mathcal{B}(H)).$$

$$B_n(Q_t)=\sum _{p\in {\mathbb{N}}^d,0\le p\le n}(-1{)}^{|p|}(\genfrac{}{}{0ex}{}{n}{p})Q_t^p(I)\le 0\mathrm{for}\mathrm{all}\mathrm{n}\in {\mathbb{N}}^{\mathrm{d}}\{0\},$$

$${{\phi }_i}_{t_i}(x)=\{\begin{array}{ll}\sqrt{\frac{{\phi }_i(x)}{{\phi }_i(x-t_i)}}& \text{if x≥ti},\\ 0& \text{if x<ti}\end{array}$$

$${S_i}_{t_i}f(x)=\{\begin{array}{ll}{{\phi }_i}_{t_i}(x)f(x-t_i)& \text{if x≥ti},\\ 0& \text{if x<ti.}\end{array}$$

$${S_i}_{t_i}{S_j}_{t_j}={S_j}_{t_j}{S_i}_{t_i}$$

$$\frac{{\phi }_i(x){\phi }_j(x-t_i)}{{\phi }_i(x-t_i){\phi }_j(x-t_i-t_j)}=\frac{{\phi }_j(x){\phi }_i(x-t_j)}{{\phi }_j(x-t_j){\phi }_i(x-t_i-t_j)}$$

$${S_i}_{t_i}^{\ast }f(x)=\sqrt{\frac{{\phi }_i(x+t_i)}{{\phi }_i(x)}}f(x+t_i)\text{mbox}foralmosteveryx\in {\mathbb{R}}_{+}.$$

$${S_i}_{t_i}^{\ast }{S_i}_{t_i}f(x)=\frac{{\phi }_i(x+t_i)}{{\phi }_i(x)}f(x)\text{mbox}foralmosteveryx\in {\mathbb{R}}_{+}.$$

$$I-{S_i}_{t_i}^{\ast }{S_i}_{t_i}=0\mathrm{for}\mathrm{all}1\le i\le d.$$

$${\phi }_i(x)-{\phi }_i(x+t_i)=0\mathrm{for}\mathrm{all}x,t_i\in {\mathbb{R}}_{+},1\le i\le d.$$

$${U_i}_{t_i}f(x)=\{\begin{array}{ll}f(x-t_i)& \text{if x≥ti},\\ 0& \text{if x<ti}\end{array}$$

$${D_i}_{t_i}f(x)=\sqrt{\frac{{\phi }_i(x+t_i)}{{\phi }_i(x)}}f(x),\forall x\ge 0.$$

$$S_t^{\prime }f(x)=\{\begin{array}{ll}\frac{1}{{\phi }_t(x)}f(x-t)& \text{if x≥t},\\ 0& \text{if x<t}.\end{array}$$

$${S_i}_{t_i}^{\prime }f(x)=\{\begin{array}{ll}\frac{1}{{{\phi }_i}_{t_i}(x)}f(x-t_i)& \text{if x≥ti},\\ 0& \text{if x<ti.}\end{array}$$

$$\sqrt{\frac{\phi (x)}{\phi (x-kt)}}=\sqrt{\frac{{\phi }_d(x)}{{\phi }_d(x-k_d{t}_d)}}\cdots \sqrt{\frac{{\phi }_1(x-k_2{t}_2-\cdots -k_d{t}_d)}{{\phi }_1(x-k_1{t}_1-\cdots -k_d{t}_d)}}.$$

$$({S_d}_{t_d}^{k_d}\cdots {S_1}_{t_1}^{k_1}f)(x)=\sqrt{\frac{\phi (x)}{\phi (x-kt)}}f(x-k_1{t}_1-\cdots -k_d{t}_d)$$

$$({S_1}_{t_1}^{k_1}\cdots {S_d}_{t_d}^{k_d}f)(x)=0\mathrm{if}x<k_1{t}_1+\cdots +k_d{t}_d.$$

$$\bigcap _{(k_1,\cdots ,k_d)\in {\mathbb{N}}^d}{S_1}_{t_1}^{k_1}\cdots {S_d}_{t_d}^{k_d}{L}^2({\mathbb{R}}_{+})\subseteq \bigcap _{(k_1,\cdots ,k_d)\in {\mathbb{N}}^d}{\chi }_{[k_1{t}_1+\cdots +k_d{t}_d,\infty )}{L}^2({\mathbb{R}}_{+}).$$

$$\bigcap _{(k_1,\cdots ,k_d)\in {\mathbb{N}}^d}{\chi }_{[k_1{t}_1+\cdots +k_d{t}_d,\infty )}{L}^2({\mathbb{R}}_{+})=\{0\}.$$

$$(k_{\alpha })=\{\begin{array}{ll}{W}_{\alpha -{ϵ}_j}^{(j)}{h}_{\alpha -{ϵ}_j}& \text{if αj≥1},\\ 0& \text{if αj=0}\end{array}$$

$${T^{\prime }}_{[j]}^{\alpha }=\{\begin{array}{ll}I& \text{if d=1},\\ {\prod }_{i\ne j}{T_i^{\prime }}^{{\alpha }_i}& \text{if d≥2.}\end{array}$$

$$r:=(r({S_1}_{t_1}^{\prime }{)}^{-1},\cdots ,r({S_d}_{t_d}^{\prime }{)}^{-1}),$$

$$U_f(z):=\sum _{k\in {\mathbb{N}}^d}(P{S_{\mathbf{t}}^{\prime }}^{\ast k}f)z^k,z\in {\mathbb{C}}^d,$$

$$(U{S_j}_{t_j})(f)(z)$$

$$k_{\mathcal{H}}(z,\lambda )e(x)=\left(\sum _{n\in {\mathbb{N}}^2}P{S_{\mathbf{t}}^{\prime }}^{\ast n}{S_{\mathbf{t}}^{\prime }}^n{z}^n{\overline{\lambda }}^n\right)e(x)=\left(\sum _{(n_1,n_2)\in {\mathbb{N}}^2}{z}_1^{n_1}{z}_2^{n_2}{\overline{{\lambda }_1}}^{n_1}{\overline{{\lambda }_2}}^{n_2}\right)e(x).$$

$$k_{\mathcal{H}}(z,\lambda )e(x)$$

$$(M_{\theta }f)(x)=e^{i\theta x}f(x).$$

Full text

1. Introduction

The class of weighted shift operators has been systematically studied in [12] and generalized to multivariable set up in [9]. With a view to study a continuous analogue of weighted shifts, M. Embry and A. Lambert initiated the study of a semigroup of operators $\{S_{t}\}$ indexed by a non-negative real number $t$ in [7], [8] and termed it as weighted translation semigroup. The operators $S_{t}$ are defined on $L^{2}(\mathbb{R}_{+})$ by using a weight function. This work is continued in [10], where we characterized some special types of weighted translation semigroups, especially, hyperexpansive weighted translation semigroups. In [11], we proved that the weighted translation semigroup $\{S_{t}\}$ is analytic and possesses wandering subspace property. Also we proved that a left invertible operator $S_{t}$ is modeled as a multiplication by $z$ on a suitable reproducing kernel Hilbert space. It turned out that the spectrum of a left invertible operator $S_{t}$ is a closed disc and the point spectrum is empty.

In section 2, we introduce the commuting tuple $S_{\bf t}$ whose components are weighted translation semigroups in ${\cal B}(L^{2}({\mathbb{R}_{+}}))$ and discuss its properties. We present several examples in this section.

In section 3, we prove that the tuple $S_{\bf t}$ is analytic and possesses wandering subspace property. We also prove that the tuple $S_{\bf t}$ is unitarily equivalent to a commuting operator valued multishift. Further, we describe the toral analytic model for the toral left invertible tuple $S_{\bf t}$. At the end of this section, the Taylor spectrum of the toral left invertible tuple $S_{\bf t}$ is discussed. The major part of the work in section 3 is devoted to the comparison of the commuting $d$-tuple $S_{\bf t}$ under consideration and a multishift $S_{\bf\lambda}$ on directed cartesian product of rooted directed trees as described in [5]. At the end of this paper, a comparative analysis of these two classes of operator tuples has been summarized. The techniques used in the proofs of results in section 3 are similar to those in [5]. However, the intrinsic differences in the classes of operators get reflected in the proofs accounting for some subtle differences. In the recent work in [3], S. Chavan developed toral and spherical analytic models for a commuting tuple of operators.

In section 4, we define the weighted translation semigroup $\{S_{\overline{t}}\}$ in ${\cal B}(L^{2}({\mathbb{R}_{+}^{d}}))$ and discuss its properties. In fact, the weighted translation semigroup $\{S_{t}\}$ is a special case where $d=1.$ In section 5, we introduce the special commuting tuple whose components are weighted translation semigroups in ${\cal B}(L^{2}({\mathbb{R}_{+}^{d}}))$ and discuss several properties. In this case, we observe that the symbols of the semigroups under consideration are multi-variable functions. This consideration results in some differences in properties of weighted translation semigroups studied earlier.

2. The Commuting tuple

2.1. Prelude

Let $T=(T_{1},\cdots,T_{d})$ be a tuple of commuting bounded linear operators $T_{i}(1\leq i\leq d)$ on a Hilbert space $H.$ Then $T^{*}$ represents $(T_{1}^{*},\cdots,T_{d}^{*})$ and for $p=(p_{1},\cdots,p_{d})\in\mathbb{N}^{d},~{}T^{p}$ denotes $T_{1}^{p_{1}}\cdots T_{d}^{p_{d}}.$ The open polydisc centered at the origin and of polyradius $r=(r_{1},\cdots,r_{d})$ with $r_{1},\cdots,r_{d}>0$ denoted by $\mathbb{D}_{r}^{d}$ is $\mathbb{D}_{r}^{d}:=\{z=(z_{1},\cdots,z_{d})\in\mathbb{C}^{d}~{}:~{}|z_{1}|<r_{1},\cdots,|z_{d}|<r_{d}\}.$ The class of completely hyperexpansive operators is explored in [14] and generalised to multi variable set up in [2]. We first recall some definitions useful in the sequel ([1],[4]).

Definition 2.1 :

1. (1)

A $d$-tuple $S=(S_{1},\cdots,S_{d})$ of commuting operators $S_{i}$ in ${\cal B}(H)$ is subnormal if there exist a Hilbert space $K$ containing $H$ and a $d$-tuple $N=(N_{1},\cdots,N_{d})$ of commuting normal operators $N_{i}$ in ${\cal B}(K)$ such that $N_{i}H\subseteq H$ and $N_{i}|H=S_{i}$ for $1\leq i\leq d.$ 2. (2)

Let $H_{1},\cdots,H_{d}$ be commuting operators on $H.$ The tuple $(H_{1},\cdots,H_{d})$ is called hyponormal if the $d\times d$ operator matrix $([H_{j}^{*},H_{i}])\geq 0,$ where

[TABLE] 3. (3)

A commuting $d$-tuple $Q=(Q_{1},\cdots,Q_{d})$ of positive operators $Q_{1},\cdots,Q_{d}$ in ${\cal B}(H)$ is called as the generating $d$-tuple on $H.$ 4. (4)

For a commuting $d$-tuple $T=(T_{1},\cdots,T_{d})$ of operators on $H,$ the spherical generating $1$-tuple associated with $T$ is given by

[TABLE] 5. (5)

For a fixed integer $p\geq 1,$ the tuple $T$ is said to be spherical $p$-expansion (resp. spherical $p$-contraction) if

[TABLE]

where $Q_{s}^{0}(I)=I.$ 6. (6)

The tuple $T$ is called spherical $p$-hyperexpansion (resp. spherical $p$-hypercontraction) if $T$ is a spherical $k$-expansion (resp. spherical $k$-contraction) for all $k=1,\cdots,p.$ If $B_{p}(Q_{s})=0,$ then $T$ is a spherical $p$-isometry. 7. (7)

The tuple $T$ is said to be spherical complete hyperexpansion (resp. spherical complete hypercontraction) if $T$ is a spherical $p$-expansion (resp. spherical $p$-contraction) for all positive integers $p.$ 8. (8)

Let $Q_{s}$ be the spherical generating $1$-tuple associated with $T.$ Then $T$ is jointly left-invertible if there exists $\alpha>0$ such that $Q_{s}(I)\geq\alpha I.$ Let $T$ be a jointly left-invertible $d$-tuple of bounded operators on $H.$ The spherical Cauchy dual of $T$ is defined as the $d$-tuple $T^{s}=(T_{1}^{s},\cdots,T_{d}^{s}),$ where $T_{i}^{s}=T_{i}(Q_{s}(I))^{-1}(i=1,\cdots,d).$ 9. (9)

Given a commuting $d$-tuple $T=(T_{1},\cdots,T_{d})$ on $H,$ a toral generating $d$-tuple is given by

[TABLE] 10. (10)

Let $Q_{t}$ be the toral generating $d$-tuple associated with $T.$ Then $T$ is toral left-invertible if there exists $\alpha>0$ such that $Q_{i}(I)\geq\alpha I,$ for $1\leq i\leq d.$ Let $T$ be a toral left-invertible $d$-tuple of bounded operators on $H.$ The toral Cauchy dual of $T$ is defined as the $d$-tuple $T^{\prime}=(T_{1}^{\prime},\cdots,T_{d}^{\prime}),$ where $T_{i}^{\prime}=T_{i}(T_{i}^{*}T_{i})^{-1}(i=1,\cdots,d).$ 11. (11)

The tuple $T$ is called toral complete hyperexpansion if

[TABLE]

where $Q_{t}^{q}(I)=(Q_{1}^{q_{1}}\circ\cdots\circ Q_{d}^{q_{d}})(I)$ for $q=(q_{1},\cdots,q_{d})\in\mathbb{N}^{d}.$

In definitions (5), (6) and (7), if $p=1$ then by convention, the prefix $1$ is dropped and if $m=1$ then we drop the term spherical.

We now define the operator $S_{t}$ as defined in [10], but in a notation suitable to multi-variable set up. For a fixed positive integer $i,$ let $\varphi_{i}$ be a measurable, positive function on $\mathbb{R}_{+}$ such that for each fixed $t_{i}\in\mathbb{R}_{+},$ the function ${\varphi_{i}}_{t_{i}}$ defined by

[TABLE]

is essentially bounded.

Definition 2.2 :

For a fixed positive integer $i$ and each fixed $t_{i}\in\mathbb{R}_{+},$ we define ${S_{i}}_{t_{i}}$ on $L^{2}({\mathbb{R}_{+}})$ by

[TABLE]

The family $G_{i}=\{{S_{i}}_{t_{i}}:t_{i}\in\mathbb{R}_{+}\}$ in ${\cal B}(L^{2}({\mathbb{R}_{+}}))$ is a semigroup with ${S_{i}}_{0}=I,$ the identity operator and for all $t_{i},r_{i}~{}\in\mathbb{R}_{+}$, ${S_{i}}_{t_{i}}\circ{S_{i}}_{r_{i}}={S_{i}}_{t_{i}+r_{i}}.$ Observe that the family $G_{1}\times\cdots\times G_{d}$ is also a semigroup.

We say that ${\varphi_{i}}_{t_{i}}$ is a weight function corresponding to the operator ${S_{i}}_{t_{i}}$. Further, the semigroup $G_{i}=\{{S_{i}}_{t_{i}}:t_{i}\in\mathbb{R}_{+}\}$ is referred to as the weighted translation semigroup with symbol $\varphi_{i}$. Throughout this article, we assume that the symbol $\varphi_{i},1\leq i\leq d$ is a continuous function on $\mathbb{R}_{+}.$

Remark 2.3 :

Consider a tuple of operators $({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}}).$ It can be seen that for $1\leq i,j\leq d,$

[TABLE]

if and only if

[TABLE]

for all $x\geq t_{i}+t_{j}.$ Observe that if $\varphi_{i}=\varphi_{j},$ then the operators ${S_{i}}_{t_{i}}$ and ${S_{j}}_{t_{j}}$ commute.

Example 2.4 :

The pair $({S_{1}}_{t_{1}},{S_{2}}_{t_{2}})$ is commuting for the following symbols:

1. (1)

$\varphi_{1}(x)=c,~{}~{}\varphi_{2}(x)=e^{x}$ 2. (2)

$\varphi_{1}(x,y)=e^{-x},~{}~{}\varphi_{2}(x,y)=e^{x}$

Recall that

[TABLE]

Also

[TABLE]

Note that the semigroup $G_{1}\times\cdots\times G_{d}$ is a toral isometry if every commuting $d$-tuple $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ in $G_{1}\times\cdots\times G_{d}$ is a toral isometry.

Remark 2.5 :

The commuting $d$-tuple $({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ is a toral isometry if and only if

[TABLE]

This is true if and only if

[TABLE]

Therefore the semigroup $G_{1}\times\cdots\times G_{d}$ is a toral isometry if and only if $\varphi_{i}$ is a constant function for $1\leq i\leq d.$

Using the characterization of a toral isometry as stated above, we prove that the toral left invertible commuting $d$-tuple $S_{\bf t}$ admits a polar decomposition.

Proposition 2.6**.**

If $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ is a toral left invertible commuting $d$-tuple, then there exist a toral isometry $({U_{1}}_{t_{1}},\cdots,{U_{d}}_{t_{d}})$ and a commuting $d$-tuple $({D_{1}}_{t_{1}},\cdots,{D_{d}}_{t_{d}})$ of diagonal, positive, invertible bounded operators on $L^{2}(\mathbb{R}_{+})$ such that ${S_{i}}_{t_{i}}={U_{i}}_{t_{i}}{D_{i}}_{t_{i}},~{}~{}~{}~{}1\leq i\leq d.$ Further, this decomposition is unique.

Proof.

For a fixed positive integer $i,~{}1\leq i\leq d,$ define the operators ${U_{i}}_{t_{i}}$ and ${D_{i}}_{t_{i}}$ on $L^{2}(\mathbb{R}_{+})$ as follows:

[TABLE]

and

[TABLE]

It can be seen that ${S_{i}}_{t_{i}}={U_{i}}_{t_{i}}{D_{i}}_{t_{i}},~{}~{}~{}~{}1\leq i\leq d$ and this decomposition is unique. ∎

We now quote the proposition which is useful in constructing examples of spherical hyperexpansion using toral hyperexpansion [4, Proposition 3.7].

Proposition 2.7**.**

Let $T=(T_{1},\cdots,T_{m})$ be an m-tuple on a Hilbert space $\cal H$ and set $S=(T_{1}/{\sqrt{m}},\cdots,T_{m}/{\sqrt{m}}).$ If $T$ is a toral complete hyperexpansion (resp. toral p-expansion, toral p-isometry) then $S$ is a spherical complete hyperexpansion (resp. spherical p-expansion, spherical p-isometry).

Note that if $T$ is a complete hyperexpansion (resp. $p$-isometry) then the $m$-tuple $(T,\cdots,T)$ is a toral complete hyperexpansion (resp. toral $p$-isometry). By Propostion 2.7, $(T/\sqrt{m},\cdots,T/\sqrt{m})$ is a spherical complete hyperexpansion (resp. spherical $p$-isometry).

We here quote proposition useful in constructing hyponormal tuples.

Proposition 2.8**.**

[1, Proposition 3]** Let $S\in\cal B(H).$ Then $S$ is subnormal if and only if $(I,S,\cdots,S^{p-1})$ is hyponormal for every $p\geq 1.$

Example 2.9 :

We here present examples of some special tuples.

1. (1)

Recall that functions $\varphi_{1}(x)=\log(x+2),\varphi_{2}(x)=\frac{x+\lambda}{x+1}(0<\lambda<1),$

$\varphi_{3}(x)=2-e^{-x}$ are completely alternating. Therefore each operator ${S_{i}}_{t_{i}}$ in a weighted translation semigroup $G_{i}$ with symbol $\varphi_{i},~{}~{}1\leq i\leq 3$ is completely hyperexpansive. Hence the pair $({S_{i}}_{t_{i}},{S_{i}}_{t_{i}})$ is a toral complete hyperexpansion and the pair $({S_{i}}_{t_{i}}/\sqrt{2},{S_{i}}_{t_{i}}/\sqrt{2})$ is a spherical complete hyperexpansion for $1\leq i\leq 3.$ 2. (2)

Let $\varphi_{1}(x)=\sqrt{x+1}.$ Since the function $\varphi_{1}$ is concave, each operator ${S_{1}}_{t_{1}}$ in a weighted translation semigroup $G_{1}$ with symbol $\varphi_{1}$ is 2-hyperexpansion. Hence the pair $({S_{1}}_{t_{1}},{S_{1}}_{t_{1}})$ is a toral 2-hyperexpansion and the pair

$({S_{1}}_{t_{1}}/\sqrt{2},{S_{1}}_{t_{1}}/\sqrt{2})$ is a spherical 2-hyperexpansion. 3. (3)

Observe that each operator ${S_{i}}_{t_{i}}$ in a weighted translation semigroup $G_{i}$ with symbol $\varphi_{i},~{}~{}1\leq i\leq 2$ is an isometry, where $\varphi_{1}(x)=c,\varphi_{2}(x)=d.$ Therefore the pair $({S_{1}}_{t_{1}},{S_{2}}_{t_{2}}),$ is a toral isometry and the pair

$({S_{1}}_{t_{1}}/\sqrt{2},{S_{2}}_{t_{2}}/\sqrt{2})$ is a spherical isometry. 4. (4)

Observe that each operator ${S_{1}}_{t_{1}}$ in a weighted translation semigroup $G_{1}$ with symbol $\varphi_{1}(x,y)=x+1$ is a 2-isometry. Hence the pair $({S_{1}}_{t_{1}},{S_{1}}_{t_{1}})$ is a toral 2-isometry and the pair $({S_{1}}_{t_{1}}/\sqrt{2},{S_{1}}_{t_{1}}/\sqrt{2})$ is a spherical 2-isometry. 5. (5)

Recall that functions $\varphi_{1}(x)=\frac{1}{x+1},\varphi_{2}(x)=\frac{x+\lambda}{x+1}(\lambda>1),\varphi_{3}(x)=e^{-x}$ are completely monotone. Therefore each operator ${S_{i}}_{t_{i}}$ in a weighted translation semigroup $G_{i}$ with symbol $\varphi_{i},~{}~{}1\leq i\leq 3$ is subnormal contraction. Hence the pair $(I,{S_{i}}_{t_{i}})$ is hyponormal.

3. Analytic Model and Taylor Spectrum

In this section, we prove that a commuting $d$-tuple $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ is analytic and possesses wandering subspace property. Also a toral left invertible commuting $d$-tuple $S_{\bf t}$ is modeled as a multiplication by coordinate functions $z_{i}$ on a suitable reproducing kernel Hilbert space.

Recall that the Cauchy dual $S_{t}^{\prime}$ of a left invertible operator $S_{t}$ is given by

[TABLE]

Let $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ be a toral left invertible commuting $d$-tuple. Then the toral Cauchy dual tuple $S_{\bf t}^{\prime}=({S_{1}}_{t_{1}}^{\prime},\cdots,{S_{d}}_{t_{d}}^{\prime})$ of $S_{\bf t}$ is given by for $1\leq i\leq d,$

[TABLE]

Note that the toral Cauchy dual $d$-tuple $({S_{1}}_{t_{1}}^{\prime},\cdots,{S_{d}}_{t_{d}}^{\prime})$ is commuting if and only if the toral left invertible $d$-tuple $({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ is commuting.

We first describe the joint kernel $E$ of the commuting $d$-tuple $S_{\bf t}^{*}=({S_{1}}_{t_{1}}^{*},\cdots,{S_{d}}_{t_{d}}^{*}).$ Recall that $E=\cap_{i=1}^{d}ker{S_{i}}_{t_{i}}^{*}$ and ker ${S_{i}}_{t_{i}}^{*}=\chi_{[0,t_{i})}L^{2}(\mathbb{R}_{+})$ (Refer [11, Lemma 3.2]). Observe that $\chi_{[0,t_{i})}L^{2}(\mathbb{R}_{+})\subset\chi_{[0,t_{j})}L^{2}(\mathbb{R}_{+})~{}{\rm if}~{}t_{i}<t_{j}.$ Thus $E=\chi_{[0,t_{j})}L^{2}(\mathbb{R}_{+}),$

$~{}{\rm where}~{}t_{j}=\min\{t_{1},\cdots,t_{d}\}.$

A commuting $d$-tuple $T=(T_{1},\cdots,T_{d})$ on a Hilbert space $H$ is called analytic if $\cap_{\alpha\in\mathbb{N}^{d}}T^{\alpha}H=\{0\}.$

Proposition 3.1**.**

A commuting $d$-tuple $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ is analytic.

Proof.

Suppose $k=(k_{1},\cdots,k_{d})\in\mathbb{N}^{d}.$ For simplicity, we use notation

[TABLE]

Observe that for $f\in L^{2}(\mathbb{R}_{+})$

[TABLE]

if $x\geq k_{1}t_{1}+\cdots+k_{d}t_{d}$ and

[TABLE]

Hence

[TABLE]

The result follows from the fact that

[TABLE]

∎

The following Lemma follows in similar manner as [11, Lemma 3.5].

Lemma 3.2**.**

Let $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ be a toral left invertible commuting $d$-tuple in $G_{1}\times\cdots\times G_{d}$ and let $E$ be the joint kernel of $S_{\bf t}^{*}.$ Then the multisequence $\{S_{\bf t}^{k}E\}_{k\in\mathbb{N}^{d}}$ is mutually orthogonal.

Remark 3.3 :

Using similar argument, it can be proved that the multisequence $\{{S_{\bf t}^{\prime}}^{k}E\}_{k\in\mathbb{N}^{d}}$ is also mutually orthogonal.

We say that a commuting $d$-tuple $T=(T_{1},\cdots,T_{d})$ on a Hilbert space $H$ possesses wandering subspace property if $H=[E]_{T},$ where $E$ is the joint kernel of $T^{*}$ and $[E]_{T}=\vee_{\alpha\in\mathbb{N}^{d}}T^{\alpha}E.$ The following Proposition follows in similar manner as [11, Proposition 3.4].

Proposition 3.4**.**

A commuting $d$-tuple $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ in $G_{1}\times\cdots\times G_{d}$ possesses wandering subspace property.

Remark 3.5 :

Using same technique given in the proof of [11, Proposition 3.4], it can be proved that the commuting $d$-tuple $S_{\bf t}^{\prime}$ also possesses wandering subspace property.

We first define operator valued multishift. Let $M$ be a nonzero complex Hilbert space and let $l^{2}_{M}(\mathbb{N}^{d})$ denote the Hilbert space of square summable multisequence $\{h_{\alpha}\}_{\alpha\in\mathbb{N}^{d}}$ in $M.$ If $\{W_{\alpha}^{(j)}\}_{\alpha\in\mathbb{N}^{d}}\subseteq\cal B(M)$ for $j=1,\cdots,d,$ then the linear operator $W_{j}$ in $l^{2}_{M}(\mathbb{N}^{d})$ is defined by $W_{j}(h_{\alpha})_{\alpha\in\mathbb{N}^{d}}=(k_{\alpha})_{\alpha\in\mathbb{N}^{d}}$ for $(h_{\alpha})_{\alpha\in\mathbb{N}^{d}}\in\cal D,$ where

[TABLE]

and ${\cal D}:=\{(h_{\alpha})_{\alpha\in\mathbb{N}^{d}}\in l^{2}_{M}(\mathbb{N}^{d})~{}:~{}(k_{\alpha})_{\alpha\in\mathbb{N}^{d}}\in l^{2}_{M}(\mathbb{N}^{d})\}.$

The $d$-tuple $W=(W_{1},\cdots,W_{d})$ is called an operator valued multishift with operator weights $\{W_{\alpha}^{(j)}~{}:~{}\alpha\in\mathbb{N}^{d},j=1,\cdots,d\}$.

The proof of the following proposition follows in a similar way as given in [5, Corollary 4.1.12].

Proposition 3.6**.**

Let $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ be a toral left invertible commuting $d$-tuple in $G_{1}\times\cdots\times G_{d}.$ Let $E$ be the joint kernel of $S_{\bf t}^{*}.$ Then $S_{\bf t}$ is unitarily equivalent to a commuting operator valued multishift $W$ on $l^{2}_{E}(\mathbb{N}^{d}).$

We now define the kernel condition useful in constructing the analytic model.

Definition 3.7 :

Let $T=(T_{1},\cdots,T_{d})$ be a toral left invertible $d$-tuple on a Hilbert space $H$ and let $E$ denote the joint kernel of $T^{*}.$ Let $T^{\prime}$ be the toral Cauchy dual of $T.$ We say that $T$ satisfies kernel condition if $E\subseteq\rm{ker}T_{j}^{*}{T^{\prime}}^{\alpha}_{[j]}$ for all $j=1,\cdots,d$ and for all $\alpha\in\mathbb{N}^{d},$ where, for $\alpha\in\mathbb{N}^{d}$ and $j=1,\cdots,d,$

[TABLE]

Note that the toral left invertible commuting $d$-tuple $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ satisfies kernel condition [5, Remark 4.2.3(iii)].

The following theorem is a multivariable analogue of Shimorin’s analytic model developed in [11, Theorem 3.7]. The reader may compare this theorem with [5, Theorem 4.2.4] for a similar model developed in the context of multishifts on directed cartesian product of rooted directed trees.

Theorem 3.8**.**

Let $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ be a toral left invertible commuting $d$-tuple in $G_{1}\times\cdots\times G_{d}.$ Let $E$ be the joint kernel of $S_{\bf t}^{*}$ and let

[TABLE]

where $r(T)$ denotes the spectral radius of the bounded operator $T.$ Then there exist a reproducing kernel Hilbert space $\cal H$ of $E$-valued analytic functions defined on the polydisc $\mathbb{D}_{r}^{d}$ with center origin and polyradius $r$ and a unitary operator $U:L^{2}(\mathbb{R}_{+})\rightarrow\cal H$ such that $U{S_{j}}_{t_{j}}=M_{z_{j}}U$ for $j=1,\cdots,d.$

Proof.

For $f\in L^{2}(\mathbb{R}_{+}),$ define

[TABLE]

where $P$ is the orthogonal projection on $E.$ The power series $U_{f}$ converges absolutely on the polydisc $\mathbb{D}_{r}^{d}.$ Let $\cal H$ denote the complex vector space of $E$-valued analytic functions of the form $U_{f}.$ We now define a map $U:L^{2}(\mathbb{R}_{+})\rightarrow\cal H$ given by $U(f)=U_{f}.$ By definition, $U$ is onto. We now need to show that $U$ is injective.

Let $U(f)=U_{f}=0$ for some $f\in L^{2}(\mathbb{R}_{+}).$ Then $\sum_{k\in\mathbb{N}^{d}}(P{S_{\bf t}^{\prime}}^{*k}f)z^{k}=0.$ This implies that $P{S_{\bf t}^{\prime}}^{*k}f=0$ for all $k\in\mathbb{N}^{d}.$ The fact ${\rm ker}~{}{S_{j}}_{t_{j}}^{\prime*}={\rm ker}~{}{S_{j}}_{t_{j}}^{*},$ implies that the joint kernel of the tuple ${S_{\bf t}^{\prime}*}=E.$ By the wandering subspace property of the $d$-tuple $S_{\bf t}^{\prime}$ [Remark 4.6], we have $\bigvee_{k\in\mathbb{N}^{d}}{S_{\bf t}^{\prime}}^{k}=L^{2}(\mathbb{R}_{+}).$ By taking orthogonal complement on both sides, we get $\bigcap_{k\in\mathbb{N}^{d}}({S_{\bf t}^{\prime}}^{k}(E))^{\bot}=\{0\}.$ Note that $({S_{\bf t}^{\prime}}^{k}(E))^{\bot}={\rm ker}~{}P{S_{\bf t}^{\prime}}^{*k}$ for every $k\in\mathbb{N}^{d}.$ Thus $\bigcap_{k\in\mathbb{N}^{d}}{\rm ker}~{}P{S_{\bf t}^{\prime}}^{*k}=\{0\}.$ Recall that $P{S_{\bf t}^{\prime}}^{*k}f=0$ for all $k\in\mathbb{N}^{d}.$ Therefore $f\in\bigcap_{k\in\mathbb{N}^{d}}{\rm ker}~{}P{S_{\bf t}^{\prime}}^{*k}.$ This implies that $f=0.$ Hence $U$ is injective.

The inner product on $\cal H$ can be now defined as $\left\langle U_{f},U_{g}\right\rangle_{\cal H}=\left\langle f,g\right\rangle_{L^{2}(\mathbb{R}_{+})}$ for all $f,g\in L^{2}(\mathbb{R}_{+}).$ Since $U$ is a sujective isometry, it is a unitary operator. A complete vector space $\cal H$ is now a Hilbert space. For $f\in L^{2}(\mathbb{R}_{+}),$

[TABLE]

Note that by kernel condition, $\sum_{k\in\mathbb{N}^{d},k_{j}=0}(P{S_{{\bf t}[j]}^{\prime}}^{*k}{S_{j}}_{t_{j}}f)z^{k}=0.$ Since the toral Cauchy dual tuple $S_{\bf t}^{\prime}$ is commuting and ${S_{j}}_{t_{j}}^{\prime*}{S_{j}}_{t_{j}}=I,$ the sum on the right hand side of the last step is equal to $z_{j}\sum_{k\in\mathbb{N}^{d}}(P{S_{\bf t}^{\prime}}^{*k}f)z^{k}=z_{j}U_{f}(z)=M_{z_{j}}U(f)(z).$ Hence $U{S_{j}}_{t_{j}}=M_{z_{j}}U.$

Using Remark 3.3 and argument similar to [11, Theorem 3.7(ii)], it can be proved that $(P{S_{\bf t}^{\prime}}^{*j}{S_{\bf t}^{\prime}}^{k})|_{E}=0$ for all $j\neq k$ in $\mathbb{N}^{d}.$ We now compute the reproducing kernel for the RKHS $\cal H$ associated to the operator tuple $S_{\bf t}.$ These computations are similar as given in the proof of [13, Proposition 2.13]. Note that $E=$ker $S_{\bf t}^{*}$ is infinite dimensional. It is easy to see that for $e\in E,~{}z,\lambda\in\mathbb{D}_{r}^{d}$ and $x\in\mathbb{R}_{+},~{}\displaystyle k_{\cal H}(z,\lambda)e(x)=\left(\sum_{n\in\mathbb{N}^{d}}P{S_{\bf t}^{\prime}}^{*n}{S_{\bf t}^{\prime}}^{n}z^{n}\overline{\lambda}^{n}\right)e(x).$ It can be seen that for $f,g\in E$ and $\lambda\in\mathbb{D}_{r}^{d},$ $\left\langle U_{f},k_{\cal H}(.,\lambda)g\right\rangle_{\cal H}=\left\langle U_{f}(\lambda),g\right\rangle_{E}.$ Hence $\cal H$ is a reproducing kernel Hilbert space with kernel $k.$ ∎

Example 3.9 :

We now compute the reproducing kernel for the RKHS $\cal H$ associated to the pair $({S_{1}}_{t_{1}},{S_{2}}_{t_{2}})$ in some special cases.

1. (1)

Let $\varphi_{1}(x)=a,\varphi_{2}(x)=b.$ Then the pair $({S_{1}}_{t_{1}},{S_{2}}_{t_{2}})$ is a toral isometry. For $e\in E,~{}z,\lambda\in\mathbb{D}_{r}^{2}$ and $x\in\mathbb{R}_{+},$

[TABLE] 2. (2)

Let $\varphi_{1}(x)=\varphi_{2}(x).$ For $e\in E,~{}z,\lambda\in\mathbb{D}_{r}^{2}$ and $x\in\mathbb{R}_{+},$

[TABLE]

3.1. Taylor Spectrum

For the definition and basic properties of Taylor spectrum, the reader is referred to [6].

Proposition 3.10**.**

Let $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ be a commuting $d$-tuple in $G_{1}\times\cdots\times G_{d}.$ For every $\theta\in\mathbb{R}$ there exists a unitary operator $M_{\theta}$ on $L^{2}(\mathbb{R}_{+})$ such that $M_{\theta}^{*}{S_{j}}_{t_{j}}M_{\theta}=e^{-i\theta t_{j}}{S_{j}}_{t_{j}}$ for $j=1,\cdots,d.$

Proof.

For $\theta\in\mathbb{R},$ define the map $M_{\theta}:L^{2}(\mathbb{R}_{+})\rightarrow L^{2}(\mathbb{R}_{+})$ as

[TABLE]

Then clearly $M_{\theta}$ is a unitary operator. It can be seen that the operator ${S_{j}}_{t_{j}}$ is unitarily equivalent to the operator $e^{-i\theta t_{j}}{S_{j}}_{t_{j}}.$ ∎

Remark 3.11 :

By the Proposition 3.10, the commuting $d$-tuples $({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ and $(e^{-i\theta t_{1}}{S_{1}}_{t_{1}},\cdots,e^{-i\theta t_{d}}{S_{d}}_{t_{d}})$ are unitarily equivalent. Hence the Taylor spectrum of a commuting $d$-tuple $S_{\bf t}$ has poly-circular symmetry, that is, for any $w=(w_{1},\cdots,w_{d})\in\sigma(S_{\bf t})$ and $z=(z_{1},\cdots,z_{d})\in\mathbb{T}^{d},z.w=(z_{1}w_{1},\cdots,z_{d}w_{d})\in\sigma(S_{\bf t}).$ In particular, $\sigma(S_{\bf t}^{*})=\sigma(S_{\bf t}).$

For proving the connectedness of a Taylor spectrum of a commuting $d$-tuple $S_{\bf t},$ we need a lemma. Given a positive integer $d,$ we set $H^{\oplus d}:=H\oplus\cdots\oplus H$ ($d$ times). For a commuting $d$-tuple $T=(T_{1},\cdots,T_{d})$ of operators on a Hilbert space $H,$ consider the linear transformation $D_{T}~{}:~{}H\rightarrow H^{\oplus d}$ given by for $h\in H$

[TABLE]

Observe that ker $D_{T}=\rm{ker}~{}T=\cap_{j=1}^{d}\rm{ker}~{}T_{j}.$

Lemma 3.12**.**

Let $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ be a commuting $d$-tuple. For $i\in\mathbb{N},$ let $T^{(i)}$ denote the commuting $d$-tuple $({{S_{1}}_{t_{1}}^{*}}^{i},\cdots,{{S_{d}}_{t_{d}}^{*}}^{i}).$ Then $\cup_{i\in\mathbb{N}}\rm{ker}~{}D_{T^{(i)}}$ is dense in $L^{2}(\mathbb{R}_{+}).$

Proof.

Note that ker $D_{T^{(i)}}=\chi_{[0,it_{r})}L^{2}(\mathbb{R}_{+}),$ where $t_{r}=\min\{t_{1},\cdots,t_{d}\}.$ It is sufficient to prove that $\cup_{i\in\mathbb{N}}\rm{ker}~{}D_{T^{(i)}}$ contains some orthonormal basis of $L^{2}(\mathbb{R}_{+}).$ Let $\{\widetilde{\psi_{jk}}\},$ where $j$ is an integer and $k$ is a non-negative integer be an orthonormal basis of $L^{2}(\mathbb{R}_{+})$ as described in [11, Theorem 3.7(iv)]. For fixed $j$ and $k,$ if $\frac{k+1}{2^{j}}<it_{r},$ then $\widetilde{\psi_{jk}}\in\rm{ker}~{}D_{T^{(i)}}.$ Therefore each $\widetilde{\psi_{jk}}\in\rm{ker}~{}D_{T^{(i)}}$ for some non-negative integer $i.$ Hence the orthonormal basis $\{\widetilde{\psi_{jk}}\}\subseteq\cup_{i\in\mathbb{N}}\rm{ker}D_{T^{(i)}}.$ ∎

Now proof of the following proposition follows on the lines similar to the proof of [5, Proposition 3.2.4].

Proposition 3.13**.**

The Taylor spectrum of a commuting $d$-tuple $S_{\bf t}=({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ is connected.

Remark 3.14 :

Recall that the point spectrum of each operator ${S_{j}}_{t_{j}}$ is empty [11, Proposition 4.3(1)]. This implies that the point spectrum of a commuting $d$-tuple $S_{\bf t}$ is empty.

Remark 3.15 :

Recall that $\mathbb{D}_{r}^{d}$ is the polydisc with center origin and polyradius $r,$ where $r=(r({S_{1}}_{t_{1}}^{\prime})^{-1},\cdots,r({S_{d}}_{t_{d}}^{\prime})^{-1}).$ Note that the polydisc $\mathbb{D}_{r}^{d}$ is contained in the point spectrum of $S_{\bf t}^{*}$ for a toral left invertible commuting $d$-tuple $S_{\bf t}.$ The proof of this fact is similar to the proof of [11, Proposition 4.3(2)]. By the poly-circular symmetry of the Taylor spectrum, $\mathbb{D}_{r}^{d}\subseteq\sigma(S_{\bf t}).$ Suppose $R$ is the polyradius $R=(r({S_{1}}_{t_{1}}),\cdots,r({S_{d}}_{t_{d}})).$ Then by the projection property of the Taylor spectrum, $\sigma(S_{\bf t})\subseteq\sigma({S_{1}}_{t_{1}})\times\cdots\times\sigma({S_{d}}_{t_{d}})=\mathbb{D}_{R}^{d}.$ Also the Taylor spectrum of $S_{\bf t}$ is connected [Proposition 3.13]. If the commuting $d$-tuple $S_{\bf t}$ is a toral isometry, then $r=R=(1,\cdots,1).$ In this case, the Taylor spectrum is a polydisc.

3.2. Comparative Analysis

We now summarize the comparative analysis of the commuting $d$-tuple $S_{\bf t}$ under consideration and a multishift $S_{\bf\lambda}$ on directed cartesian product of rooted directed trees as described in [5].

1. (1)

The joint kernel of the tuple $S_{\bf t}^{*}$ is always infinite dimensional but the joint kernel of $S_{\bf\lambda}^{*}$ may be finite dimensional [5, Corollary 3.1.16]. 2. (2)

The toral Cauchy dual tuple $S_{\bf t}^{\prime}$ is always commuting but the toral Cauchy dual tuple $S_{\bf\lambda}^{\prime}$ may not be commuting [5, Proposition 5.1.1]. 3. (3)

The tuple $S_{\bf t}$ admits polar decomposition. However, the multishift $S_{\bf\lambda}$ admits polar decomposition if and only if the toral Cauchy dual tuple of $S_{\bf\lambda}$ is commuting [5, Proposition 5.1.1]. 4. (4)

The multisequence $\{S_{\bf t}^{k}E\}_{k\in\mathbb{N}^{d}}$ is mutually orthogonal but, the multisequence $\{S_{\bf\lambda}^{k}E\}_{k\in\mathbb{N}^{d}}$ may not be mutually orthogonal. 5. (5)

The toral left invertible commuting $d$-tuple $S_{\bf t}$ satisfies kernel condition but the multishift $S_{\bf\lambda}$ may not satisfy kernel condition. (refer to paragraph after [5, Remark 4.2.3]). 6. (6)

The Taylor spectrum of the tuple $S_{\bf t}$ has poly circular symmetry. However, the operator tuple $S_{\bf\lambda}$ is strongly circular [5, Proposition 3.2.1].

4. Weighted Translation semigroups in ${\cal B}(L^{2}({\mathbb{R}_{+}^{d}}))$

All the properties of the weighted translation semigroup $\{S_{t}\}$ in ${\cal B}(L^{2}({\mathbb{R}_{+}}))$ proved in [10] and [11], are also shared by the semigroup $\{S_{\overline{t}}\}$ in ${\cal B}(L^{2}({\mathbb{R}_{+}^{d}})).$ For positive integer $d,$ let $L^{2}({\mathbb{R}_{+}^{d}})$ denote the Hilbert space of complex valued square integrable Lebesgue measurable functions on $\mathbb{R}_{+}^{d}.$ Let ${\cal B}(L^{2}({\mathbb{R}_{+}^{d}}))$ denote the algebra of bounded linear operators on $L^{2}({\mathbb{R}_{+}^{d}}).$

Definition 4.1 :

For a measurable, positive function $\varphi$ defined on $\mathbb{R}_{+}^{d}$ and

$\overline{t}=(t_{1},\cdots,t_{d}),\overline{x}=(x_{1},\cdots,x_{d})\in\mathbb{R}_{+}^{d}$ define the function

$\varphi_{\overline{t}}:\mathbb{R}_{+}^{d}\rightarrow\mathbb{R}_{+}$ by

[TABLE]

Here, $\overline{x}-\overline{t}=(x_{1}-t_{1},\cdots,x_{d}-t_{d}).$

Suppose that $\varphi_{\overline{t}}$ is essentially bounded for every $\overline{t}\in\mathbb{R}_{+}^{d}$.

Definition 4.2 :

For each fixed $\overline{t}\in\mathbb{R}_{+}^{d},$ we define $S_{\overline{t}}$ on $L^{2}({\mathbb{R}_{+}^{d}})$ by

[TABLE]

Remark 4.3 :

It is easy to see that for every $\overline{t}\in\mathbb{R}_{+}^{d},~{}S_{\overline{t}}$ is a bounded linear operator on $L^{2}({\mathbb{R}_{+}^{d}})$ with $\|S_{\overline{t}}\|=\|\varphi_{\overline{t}}\|_{\infty},$ where $\|\varphi_{\overline{t}}\|_{\infty}$ stands for the essential supremum of $\varphi_{\overline{t}}.$ The family $\{S_{\overline{t}}:t\in\mathbb{R}_{+}^{d}\}$ in ${\cal B}(L^{2}({\mathbb{R}_{+}^{d}}))$ is a semigroup with $S_{\overline{0}}=I,$ the identity operator and for all $\overline{t},\overline{s}~{}\in\mathbb{R}_{+}^{d}$, $S_{\overline{t}}\circ S_{\overline{s}}=S_{\overline{t+s}}.$ Here, $\overline{0}$ is the $d$-tuple $(0,\cdots,0)$ in $\mathbb{R}_{+}^{d}.$

We say that $\varphi_{\overline{t}}$ is a weight function corresponding to the operator $S_{\overline{t}}$. Further, the semigroup $\{S_{\overline{t}}:\overline{t}\in\mathbb{R}_{+}^{d}\}$ is referred to as the weighted translation semigroup with symbol $\varphi$. Throughout this work, we assume that the symbol $\varphi$ is a continuous function on $\mathbb{R}_{+}^{d}.$

By similar computations as in the case of an operator $S_{t}$ in ${\cal B}(L^{2}(\mathbb{R}_{+})),$ it is easy to see that the adjoint of $S_{\overline{t}}$ is given by

[TABLE]

and

[TABLE]

Here, $\overline{x}+\overline{t}=(x_{1}+t_{1},\cdots,x_{d}+t_{d}).$

We now present the special types of weighed translation semigroups by choosing some special types of symbols. The characterizations of weighed translation semigroups in this case are similar to those as described in [10, Corollary 3.3].

Example 4.4 :

Let $\{S_{\overline{t}}\}$ be a weighted translation semigroup with symbol $\varphi.$

1. (1)

Let $\varphi_{1}(x,y)=e^{(x+y)},\varphi_{2}(x,y)=e^{\sqrt{xy}}.$ All these functions are log convex. Thus the semigroups $\{S_{\overline{t}}\}$ corresponding to symbols $\varphi_{i},i=1,2$ are hyponormal. 2. (2)

Let $\varphi_{1}(x,y)=2x-y-x^{2}+2xy-y^{2}+1,\varphi_{2}(x,y)=Ax^{a}y^{b}+1\\ (A>0,a,b\geq 0,a+b\leq 1).$ All these functions are concave. Thus the semigroups $\{S_{\overline{t}}\}$ corresponding to symbols $\varphi_{i},i=1,2$ are 2-hyperexpansive. 3. (3)

The semigroup $\{S_{\overline{t}}\}$ is an $m$-isometry if $\phi$ is a polynomial of degree $m-1.$ 4. (4)

Let $\varphi_{1}(x,y)=\log(x+y+2),\varphi_{2}(x,y)=\sqrt{(x+1)(y+1)},\\ \varphi_{3}(x,y)=x+y+1.$ All these functions are completely alternating. Thus the semigroups $\{S_{\overline{t}}\}$ corresponding to symbols $\varphi_{i},i=1,2,3$ are completely hyperexpansive. 5. (5)

Let $\varphi_{1}(x,y)=\frac{1}{x+y+1},~{}\varphi_{2}(x,y)=\frac{1}{\sqrt{x+y+1}},~{}\varphi_{3}(x,y)=e^{-(x+y)}.$ All these functions are completely monotone. Thus the semigroups $\{S_{\overline{t}}\}$ corresponding to symbols $\varphi_{i},i=1,2,3$ are subnormal contractions. 6. (6)

Let $\varphi_{1}(x,y)=ye^{x}+xe^{y}+1,\varphi_{2}(x,y)=e^{xy}.$ All these functions are absolutely monotone. Thus the semigroups $\{S_{\overline{t}}\}$ corresponding to symbols $\varphi_{i},i=1,2$ are alternatingly hyperexpansive. 7. (7)

Let $\varphi(x,y)=\frac{x+y+\lambda}{x+y+1},~{}\lambda\geq 0.$ Then $\frac{\partial^{n}\varphi}{\partial x^{k_{1}}\partial y^{k_{2}}}=\frac{(1-\lambda)(-1)^{n-1}n!}{(x+y+1)^{n+1}},$ where

$n=k_{1}+k_{2}.$ If $0<\lambda<1,$ then $\varphi$ is a completely alternating function and the semigroup $\{S_{\overline{t}}\}$ is completely hyperexpansive. If $\lambda>1,$ then $\varphi$ is a completely monotone function and the semigroup $\{S_{\overline{t}}\}$ is a subnormal contraction.

We now describe the kernel of the adjoint $S_{\overline{t}}^{*},$ for $\overline{t}\neq\overline{0}.$

Lemma 4.5**.**

For $\overline{t}\neq\overline{0},$ ker $S_{\overline{t}}^{*}=E=\chi_{[0,t_{1})\times\cdots\times[0,t_{d})}L^{2}(\mathbb{R}_{+}^{d}).$ In particular, ker $S_{\overline{t}}^{*}$ is infinite dimensional.

Remark 4.6 :

The following properties of $S_{\overline{t}}$ are analogues of the corresponding properties of $S_{t}.$ Let $\overline{t}\neq\overline{0}.$

1. (1)

An operator $S_{\overline{t}}$ is analytic. 2. (2)

An operator $S_{\overline{t}}$ possesses wandering subspace property. 3. (3)

A left invertible operator $S_{\overline{t}}$ possesses an analytic model. 4. (4)

A left invertible operator $S_{\overline{t}}$ is an operator valued weighted shift. 5. (5)

The spectrum of a left invertible operator $S_{\overline{t}}$ is a disc and the point spectrum is empty.

5. The Special Tuple

We now define a tuple of a special type. Recall that a commuting $d$-tuple $S_{\bf t}$ defined in Section 2, is a toral isometry if and only if each symbol $\varphi_{i},$

$1\leq i\leq d$ is a constant function [Remark 2.5]. However, in the special case, defined below we do have non-constant functions as symbols for a commuting tuple which is a toral isometry.

Let $\varphi_{i}$ be a measurable, positive function on $\mathbb{R}_{+}^{d}$ such that for each fixed $t_{i}\in\mathbb{R}_{+},$ the function ${\varphi_{i}}_{t_{i}}$ defined by

[TABLE]

is essentially bounded.

Definition 5.1 :

For each fixed $t_{i}\in\mathbb{R}_{+},$ we define ${S_{i}}_{t_{i}}$ on $L^{2}({\mathbb{R}_{+}^{d}})$ by

[TABLE]

Remark 5.2 :

It is easy to see that for every $t_{i}\in\mathbb{R}_{+},~{}{S_{i}}_{t_{i}}$ is a bounded linear operator on $L^{2}({\mathbb{R}_{+}^{d}})$ with $\|{S_{i}}_{t_{i}}\|=\|{\varphi_{i}}_{t_{i}}\|_{\infty},$ where $\|{\varphi_{i}}_{t_{i}}\|_{\infty}$ stands for the essential supremum of ${\varphi_{i}}_{t_{i}}.$ The family $G_{i}=\{{S_{i}}_{t_{i}}:t_{i}\in\mathbb{R}_{+}\}$ in ${\cal B}(L^{2}({\mathbb{R}_{+}^{d}}))$ is a semigroup with ${S_{i}}_{0}=I,$ the identity operator. Note that the family $G_{1}\times\cdots\times G_{d}$ is also a semigroup.

We say that ${\varphi_{i}}_{t_{i}}$ is a weight function corresponding to the operator ${S_{i}}_{t_{i}}$. Further, the semigroup $G_{i}=\{{S_{i}}_{t_{i}}:t_{i}\in\mathbb{R}_{+}\}$ is referred to as the weighted translation semigroup with symbol $\varphi_{i}$. Throughout this work, we assume that the symbol $\varphi_{i},1\leq i\leq d$ is a continuous function on $\mathbb{R}_{+}^{d}.$

Remark 5.3 :

Observe that ${S_{i}}_{t_{i}}=S_{\overline{t}},$ where $\overline{t}=(0,\cdots,0,t_{i},0,\cdots,0).$ Here, the $i^{th}$ component of the $d$-tuple $\overline{t}$ is $t_{i}$ and all other entries are zero. Therefore by Remark 4.6, each operator ${S_{i}}_{t_{i}}$ is analytic and possesses wandering subspace property. Also the spectrum of each left invertible operator ${S_{i}}_{t_{i}}$ is a closed disc and the point spectrum is empty.

Consider a tuple $({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})\in G_{1}\times\cdots\times G_{d}.$ It is easy to see that for $1\leq i,j\leq d,$

[TABLE]

if and only if

[TABLE]

[TABLE]

for all $x_{i}\geq t_{i},x_{j}\geq t_{j}.$ Note that if $\varphi_{i}=\varphi_{j},$ then above condition is satisfied.

Example 5.4 :

Let $b,c,\lambda$ be positive real numbers. The pair $({S_{1}}_{t_{1}},{S_{2}}_{t_{2}})$ is commuting for the following symbols:

1. (1)

$\varphi_{1}(x,y)=c,~{}~{}\varphi_{2}(x,y)=b$ 2. (2)

$\varphi_{1}(x,y)=c,~{}~{}\varphi_{2}(x,y)=e^{-x-y}$ 3. (3)

$\varphi_{1}(x,y)=c,~{}~{}\varphi_{2}(x,y)=e^{x+y}$ 4. (4)

$\varphi_{1}(x,y)=e^{-x-y},~{}~{}\varphi_{2}(x,y)=e^{x+y}$ 5. (5)

$\varphi_{1}(x,y)=y+1,~{}~{}\varphi_{2}(x,y)=x+1$ 6. (6)

$\displaystyle\varphi_{1}(x,y)=\frac{x+\lambda}{x+y+1},~{}~{}\varphi_{2}(x,y)=\frac{y+\lambda}{x+y+1}$

It is easy to see that

[TABLE]

for almost every $(x_{1},\cdots,x_{d})\in\mathbb{R}_{+}^{d}$ and

[TABLE]

for almost every $(x_{1},\cdots,x_{d})\in\mathbb{R}_{+}^{d}.$

We say that the semigroup $G_{1}\times\cdots\times G_{d}$ is a toral isometry if every commuting $d$-tuple $({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ in $G_{1}\times\cdots\times G_{d}$ is a toral isometry.

Remark 5.5 :

The commuting $d$-tuple $({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ is a toral isometry if and only if

[TABLE]

A simple calculation reveals that this condition holds if and only if

[TABLE]

Therefore the semigroup $G_{1}\times\cdots\times G_{d}$ is a toral isometry if and only if

[TABLE]

But this condition holds if and only if $\varphi_{i}$ is independent of $x_{i}$ for $1\leq i\leq d.$

Remark 5.6 :

The toral left invertible commuting $d$-tuple $({S_{1}}_{t_{1}},\cdots,{S_{d}}_{t_{d}})$ also admits polar decomposition as given in the Proposition 2.6.

We now turn our attention towards constructing examples of different classes of operator tuples. The definitions of classes of operator tuples under consideration are given in Section 2. We use Proposition 2.7 to construct following examples.

Example 5.7 :

1. (1)

The functions $\varphi_{1}(x,y)=\sqrt{x+y+1},\varphi_{1}(x,y)=\log(x+y+2),$

$\varphi_{1}(x,y)=\frac{x+y+\lambda}{x+y+1}(0<\lambda<1)$ are completely alternating. Therefore each weighted translation semigroup ${S_{1}}_{t_{1}}$ with symbol $\varphi_{1}$ is completely hyperexpansive. Hence each pair $({S_{1}}_{t_{1}},{S_{1}}_{t_{1}})$ is a toral complete hyperexpansion and the pair $({S_{1}}_{t_{1}}/\sqrt{2},{S_{1}}_{t_{1}}/\sqrt{2})$ is a spherical complete hyperexpansion. 2. (2)

The pair $({S_{1}}_{t_{1}},{S_{2}}_{t_{2}}),$ where $\varphi_{1}(x,y)=y+1,\varphi_{2}(x,y)=x+1$ is a toral isometry. Then the pair $({S_{1}}_{t_{1}}/\sqrt{2},{S_{2}}_{t_{2}}/\sqrt{2})$ is a spherical isometry. 3. (3)

The pair $({S_{1}}_{t_{1}},{S_{2}}_{t_{2}}),$ where $\varphi_{1}(x,y)=c,\varphi_{2}(x,y)=x+1$ is a toral 2-isometry. Then the pair $({S_{1}}_{t_{1}}/\sqrt{2},{S_{2}}_{t_{2}}/\sqrt{2})$ is a spherical 2-isometry. 4. (4)

The weighted translation semigroup corresponding to the symbol

$\varphi_{1}(x,y)=x+y+1$ is 2-isometry. Hence the pair $({S_{1}}_{t_{1}},{S_{1}}_{t_{1}})$ is a toral 2-isometry and the pair $({S_{1}}_{t_{1}}/\sqrt{2},{S_{1}}_{t_{1}}/\sqrt{2})$ is a spherical 2-isometry.

We here quote a proposition useful in constructing hyponormal tuples.

Proposition 5.8**.**

[1, Remark 7]** If $H_{1},\cdots,H_{m}\in{\cal B}(H),~{}~{}H_{i}^{*}H_{j}=H_{j}^{*}H_{i}$ for $i\neq j$ and each $H_{i}$ is hyponormal, then the tuple $(H_{1},\cdots,H_{m})$ is hyponormal.

Note that the above proposition is not useful to construct hyponormal tuples of the type $S_{\bf t}$ defined in Section 2.

Consider the pair $({S_{1}}_{t_{1}},{S_{2}}_{t_{2}}).$ It is easy to see that

[TABLE]

if and only if

[TABLE]

and

[TABLE]

if and only if

[TABLE]

Example 5.9 :

We now construct examples of hyponormal tuples.

1. (1)

Let $b,c$ be positive real numbers. The pair $({S_{1}}_{t_{1}},{S_{2}}_{t_{2}})$ is hyponormal for the following pair of functions:

1. (a)

$\varphi_{1}(x,y)=c,~{}~{}\varphi_{2}(x,y)=b$ 2. (b)

$\varphi_{1}(x,y)=e^{x},~{}~{}\varphi_{2}(x,y)=e^{y}$ 3. (c)

$\varphi_{1}(x,y)=c,~{}~{}\varphi_{2}(x,y)=e^{x+y}$ 4. (d)

$\varphi_{1}(x,y)=e^{-x-y},~{}~{}\varphi_{2}(x,y)=e^{x+y}$ 5. (e)

$\varphi_{1}(x,y)=e^{-y},~{}~{}\varphi_{2}(x,y)=e^{-x}$ 2. (2)

The functions $\varphi_{1}(x,y)=e^{-(x+y)},\varphi_{1}(x,y)=\frac{1}{x+y+1}$ are completely monotone. Therefore each weighted translation semigroup ${S_{1}}_{t_{1}}$ with symbol $\varphi_{1}$ is subnormal. Hence by Proposition 2.8, each tuple $(I,{S_{1}}_{t_{1}},{S_{1}}_{t_{1}}^{2},\cdots,{S_{1}}_{t_{1}}^{p-1}),$

$p\geq 1$ is hyponormal.

Using similar techniques as discussed in Section 3, the toral analytic model for the special tuple can be constructed. Further, the properties of Taylor spectrum are similar.