Operator moment dilations as block operators

TL;DR

This paper explores operator moment dilations as block operators, providing conditions for their existence and explicit block representations, extending classical moment problem results to broader classes of operators.

Contribution

It introduces a general framework for operator moment dilations, characterizes their existence, and offers explicit block operator forms for certain classes, including well-known dilations.

Findings

Block tridiagonal representations for self-adjoint dilations

Explicit block forms for $ ho$-dilations including Sch"affer and Ando representations

Identification of $ ext{C}_A$-class operators with tractable dilations

Abstract

Let $\mathcal{H}$ be a complex Hilbert space and let $\big\{A_{n}\big\}_{n\geq 1}$ be a sequence of bounded linear operators on $\mathcal{H}$. Then a bounded operator $B$ on a Hilbert space $\mathcal{K} \supseteq \mathcal{H}$ is said to be a dilation of this sequence if \begin{equation*} A_{n} = P_{\mathcal{H}}B^{n}|_{\mathcal{H}} \; \text{for all}\; n\geq 1, \end{equation*} where $P_{\mathcal{H}}$ is the projection of $\mathcal{K}$ onto $\mathcal{H}.$ The question of existence of dilation is a generalization of the classical moment problem. We recall necessary and sufficient conditions for the existence of self-adjoint, isometric and unitary dilations and present block operator representations for these dilations. For instance, for self-adjoint dilations one gets block tridiagonal representations similar to the classical moment problem. Given a positive invertible operator $A$, an operator $T$ is said to be in the $\mathcal{C}_{A}$-class if the sequence $\{A^{-\frac{1}{2}}T^nA^{-\frac{1}{2}}:n\geq 1\}$ admits a unitary dilation. We identify a tractable collection of $\mathcal{C}_A$-class operators for which isometric and unitary dilations can be written down explicitly in block operator form. This includes the well-known $\rho$-dilations for positive scalars. Here the special cases $\rho =1$ and $\rho =2$ correspond to Sch\"{a}ffer representation for contractions and Ando representation for operators with numerical radius not more than one respectively.

Equations

$$A_n=P_{\mathcal{H}}{B}^n{|}_{\mathcal{H}}\text{for all}n\ge 1,$$

$$A_n=P_{\mathcal{H}}{B}^n{|}_{\mathcal{H}}\text{for all}n\ge 1.$$

$$A_n=⟨u,B^{n}u⟩={\int }_{\sigma (B)}{x}^{n}d\mu (x)$$

$$A_{n}=P_{\mathcal{H}}B^{n}\big{|}_{\mathcal{H}},\;\text{~{}for all~{}}\;n\geq 0.\;$$

$$\mathcal{K}=\overline{\mathrm{span}}\{B^n(\mathcal{H}):n\in {\mathbb{Z}}_{+}\}.$$

$$\mathcal{K}=\overline{\mathrm{span}}\{B^n(\mathcal{H}):n\in \mathbb{Z}\}.$$

$$m_n={\int }_K{x}^{n}d\mu (x),\forall n\ge 0.$$

$$m_n=\underset{-\infty }{\overset{\infty }{\int }}{x}^{n}d\mu (x).$$

$$H_n$$

$$H_n\ge 0\text{ for every }n\ge 0.$$

$$H_n\ge 0\text{ and }{H}_n^{(1)}\ge 0\text{ for every }n\ge 0.$$

$$(-1{)}^k({\Delta }^{k}m{)}_n\ge 0\text{ for every }n,k\ge 0,$$

$$({\Delta }^{k}m{)}_n=\sum _{i=0}^k(\genfrac{}{}{0ex}{}{k}{i})(-1{)}^i{m}_{i+n}.$$

$$\mathcal{K}:=\mathcal{H}\oplus {\mathcal{D}}_T\oplus {\mathcal{D}}_T\oplus \cdots .$$

$$V=\left[\begin{array}{cccc}T& 0& 0& \cdots \\ D_T& 0& 0& \cdots \\ 0& I& 0& \cdots \\ 0& 0& I& \cdots \\ ⋮& ⋮& \ddots & \ddots \end{array}\right]$$

$$U=\left(\begin{array}[]{ccccccc}&&&&&&\\ \ddots&&&&&&\\ &I&&&&&\\ &&I&&&&\\ &&&D_{T^{*}}&{\bf T}&&\\ &&&-T^{*}&D_{T}&&\\ &&&&&I&\\ &&&&&&\ddots\end{array}\right).$$

$$c_0+c_1{A}_1+\cdots +c_n{A}_n\ge 0,$$

$$⟨g,(c_0+c_1{A}_1+c_2{A}_n+\cdots +c_n{A}_n)g⟩=⟨g,(c_0+c_{1}B+c_2{B}^2+\cdots +c_n{B}^n)g⟩\ge 0,$$

$$\phi (x^n)=A_n,\text{ for all }n\ge 0,$$

$$\phi (a)=V^{\ast }\pi (a)V,\text{for all}a\in \mathcal{A}.$$

$$A_{n}=\varphi(x^{n})=V^{\ast}\pi(x^{n})V=V^{\ast}\pi(x)^{n}V=V^{\ast}B^{n}V=P_{V(\mathcal{H})}B^{n}\big{|}_{V(\mathcal{H})},~{}~{}\text{ for all }n\geq 0.$$

$$\displaystyle A_{2}-A_{1}^{2}=\varphi(x^{2})-\varphi(x)^{2}=V^{\ast}\pi(x)\big{[}I-VV^{\ast}\big{]}\pi(x)V\geq 0,$$

$$H_n:=\left[\begin{array}{ccccc}I& A_1& A_2& \cdots & A_n\\ A_1& A_2& A_3& \cdots & A_{n+1}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ A_n& A_{n+1}& A_{n+2}& \cdots & A_{2n}\end{array}\right]\ge 0,\text{for all}n\ge 0.$$

$$\displaystyle\big{\langle}g,\;\Big{(}\sum\limits_{i,j=0}^{n}X_{i}^{\ast}A_{i+j}X_{j}\Big{)}g\big{\rangle}$$

$$\displaystyle\Big{\langle}\begin{bmatrix}h_{0}\\ \vdots\\ h_{n}\end{bmatrix},\;H_{n}\begin{bmatrix}h_{0}\\ \vdots\\ h_{n}\end{bmatrix}\Big{\rangle}$$

$$\displaystyle=\Big{\langle}g,\;\Big{(}\sum\limits_{i,j=0}^{n}|h_{i}\rangle\langle g|^{\ast}A_{i+j}|h_{j}\rangle\langle g|\Big{)}g\Big{\rangle}$$

$$=⟨g,\sum _{i,j=0}^n{X}_i^{\ast }{A}_{i+j}{X}_{j}g⟩$$

$$\ge 0.$$

$$H_n:=\left[\begin{array}{ccccc}I& A_1& A_2& \cdots & A_n\\ A_1& A_2& A_3& \cdots & A_{n+1}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ A_n& A_{n+1}& A_{n+2}& \cdots & A_{2n}\end{array}\right],H_n^{(2)}:=\left[\begin{array}{ccccc}{A}_2& A_3& A_4& \cdots & A_{n+2}\\ A_3& A_4& A_5& \cdots & A_{n+3}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ A_{n+2}& A_{n+3}& A_{n+4}& \cdots & A_{2n+2}\end{array}\right].$$

$$L=\left[\begin{array}{ccccc}1& x& x^2& \cdots & x^n\\ 0& 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ 0& 0& 0& \cdots & 0\end{array}\right].$$

$$G=\left[\begin{array}{cccc}\sqrt{1-x^2}& x\sqrt{1-x^2}& \cdots & x^n\sqrt{1-x^2}\\ 0& 0& \cdots & 0\\ 0& 0& \cdots & 0\\ ⋮& ⋮& \cdots & ⋮\\ 0& 0& \cdots & 0\end{array}\right].$$

$$k((m,g),(n,h))=⟨g,A_{m+n}h⟩,\text{for every}m,n\in {\mathbb{Z}}_{+},g,h\in \mathcal{H}.$$

Full text

[TABLE]

where $P_{\mathcal{H}}$ is the projection of $\mathcal{K}$ onto $\mathcal{H}.$ The question of existence of dilation is a generalization of the classical moment problem. We recall necessary and sufficient conditions for the existence of self-adjoint, isometric and unitary dilations and present block operator representations for these dilations. For instance, for self-adjoint dilations one gets block tridiagonal representations similar to the classical moment problem.

Given a positive invertible operator $A$, an operator $T$ is said to be in the $\mathcal{C}_{A}$-class if the sequence $\{A^{-\frac{1}{2}}T^{n}A^{-\frac{1}{2}}:n\geq 1\}$ admits a unitary dilation. We identify a tractable collection of $\mathcal{C}_{A}$-class operators for which isometric and unitary dilations can be written down explicitly in block operator form. This includes the well-known $\rho$-dilations for positive scalars. Here the special cases $\rho=1$ and $\rho=2$ correspond to Schäffer representation for contractions and Ando representation for operators with numerical radius not more than one respectively.

1. Introduction

Starting from the pioneering work of Sz.-Nagy ([25]), the dilation theory has played a fundamental role in operator theory. This has been explored by many. The early works mostly used techniques from classical function theory and basic operator theory. Thanks to W. Arveson, V. Paulsen, and others ([28]) we have new tools coming from the theory of completely positive maps. The literature here is vast and so we simply refer to a recent survey of dilation theory by Orr Shalit [35] and the references therein.

Let $\mathcal{H}$ be a complex Hilbert space and let \big{\{}A_{n}\big{\}}_{n\geq 1} be a sequence of bounded linear operators on $\mathcal{H}$. Then a bounded operator $B$ on a Hilbert space $\mathcal{K}\supseteq\mathcal{H}$ is said to be a dilation if

[TABLE]

Generally it is convenient to include $n=0$ in the sequence, where we would be taking $A_{0}=I$ (the identity operator of the Hilbert space). The operator moment problem is to determine conditions on the sequence of operators to ensure existence of a dilation $B$ with prescribed property, for instance we may demand $B$ to be unitary/isometry/self-adjoint/positive or we may require the spectrum of $B$ to be contained in a given subset of complex plane.

Two special cases of this problem are very well-known. A famous result of Sz.-Nagy tells us that if $A_{n}=A^{n},n\geq 0$ for some operator $A$, then it admits a unitary dilation if and only if $A$ is a contraction.

Another special situation is when ${\mathcal{H}}$ is one dimensional so that $A_{n}$ is a sequence of scalars and $B$ is a self-adjoint operator on ${\mathcal{K}}$. This amounts to requiring

[TABLE]

where $u$ is a unit vector in ${\mathcal{K}}$ and $\mu$ is the spectral measure of $B$ coming from the vector state $\langle u,(\cdot)u\rangle.$ In other words, this is the classical moment problem of seeking a measure with specified moments.

The general operator moment problem is less well-known, but if we start searching we find considerable literature scattered here and there. This problem was already thought of by Sz.-Nagy ([26]). Some further references can be found in [13, 21, 39]). Also, we refer an excellent book of V. I. Paulsen [28], where the author discusses the dilation problem for operator valued sequences, mostly through various exercises.

One of the main purpose of this article is to present some of the basic results in the field in a unified way. Some of the early literature mentioned above were inspired by classical moment problems and typically use similar methods. Here in we mostly use modern techniques coming from the theory of completely positive maps or we use the method of positive kernels to show the existence of the dilation. Another aspect we emphasize is the existence of dilations as block operator matrices, the kind got by Schäffer for Sz.-Nagy dilation. This seems to be a fairly general phenomenon. This is very useful as it helps us to visualize the dilation. For the powers of a contraction, the Schäffer construction gives the dilation as a $2\times 2$ block operator perturbation of the bilateral shift. In the last Section we have $4\times 4$ block operator perturbations of the bilateral shift appearing as dilation operators.

Throughout the article, $\mathcal{H}$ denotes a complex Hilbert space and we follow Physicists convention by considering the inner product $\langle\cdot,\cdot\rangle$ on $\mathcal{H}$ as linear in the second variable and anti-linear in the first variable. Also note that $\mathcal{B}(\mathcal{H})$ denotes the space of all bounded linear operators on $\mathcal{H}$ and $C(X)$ is the $C^{\ast}$-algebra of all continuous functions on a compact Hausdorff space $X.$ In general, we denote $C^{\ast}$-algebras by $\mathcal{A},\mathcal{B}$ etc. In particular, the $C^{\ast}$ algebra of all $n\times n$ matrices with entries from $\mathcal{A}$ is denoted by $M_{n}(\mathcal{A})$. For a subset $M$ of $\mathcal{H}$, the subspace $[M]:=\overline{\rm span}(M)$ is the smallest closed subspace of $\mathcal{H}$ containing $M$.

Definition 1.1** (Dilation of operator sequences).**

Let $\{A_{n}\}_{n\geq 0}$ be a sequence of bounded linear operators on a Hilbert space $\mathcal{H}$ with $A_{0}=I$. The operator sequence $\{A_{n}\}_{n\geq 0}$ is said to admit a dilation if there exist a bounded linear operator $B$ on a Hilbert space $\mathcal{K}\supseteq\mathcal{H}$ such that

[TABLE]

*Then $B$ is called the dilation. A dilation is said to be positive/self-adjoint/isometry/unitary if $B$ has that property. A positive/self-adjoint/isometric dilation is said to be minimal if *

[TABLE]

Some clarifications are in order. Here by convention for any operator $B$, $B^{0}$ is taken as identity. So the condition A_{0}=P_{\mathcal{H}}B^{0}\big{|}_{\mathcal{H}} is superfluous once we take $A_{0}=I.$ Therefore we are effectively just considering the dilation of $\{A_{n}\}_{n\in{\mathbb{N}}}$. However, for minimality it is important to include $n=0$ in Equation (2). In the case of unitary dilations, the minimality condition in Equation (2) should be replaced by

[TABLE]

Problem 1.2**.**

Given a sequence of operators we wish to consider dilations with prescribed property of being positive, self-adjoint etc. Some natural questions that arise are the following:

1. (1)

Existence:* What are necessary and sufficient conditions for existence of dilations with prescribed property?* 2. (2)

Uniquenss:* When a dilation with prescribed property exists, is it possible to prove uniqueness of dilation up to unitary equivalence under the assumption of minimality?* 3. (3)

Construction:* Can we explicitly construct these dilations instead of just abstractly proving their existence and uniqueness.*

The question about uniqueness is easy to answer. For easy reference we state it as a theorem.

Theorem 1.3**.**

Let $\{A_{n}\}_{n\geq 0}$ be a sequence of bounded operators on a Hilbert space $\mathcal{H}$ admitting a self-adjoint/positive/isometric/unitary dilation $B$ on a Hilbert space $\mathcal{K}\supseteq\mathcal{H}.$ Then the given operator sequence admits a minimal dilation with the same prescribed property. Moreover, such a minimal dilation is unique up to unitary equivalence.

Proof.

Suppose $B$ is a self-adjoint dilation. Then $B$ restricted to $\overline{\mbox{span}}\{B^{n}h:h\in\mathcal{H},n\in\mathbb{Z}_{+}\}$ is again a self-adjoint dilation. Moreover, the inner products, $\langle B^{m}g,B^{n}h\rangle,g,h\in\mathcal{H},m,n\in\mathbb{Z}_{+}$ are completely determined by the given sequence $\{A^{n}:n\in\mathbb{Z}_{+}\}.$ This shows the uniqueness of minimal dilation up to unitary equivalence. Clearly the same statement holds for positive and isometric dilations. For unitary dilations we change $\mathbb{Z}_{+}$ by $\mathbb{Z}$ and we have the analogous result. ∎

The first question is more delicate. It is obvious that given an arbitrary operator valued sequence $\{A_{n}\}_{n\geq 0},$ there may not exist $B$ such that dilation Equation (1) holds. Some necessary conditions follow easily. We list the following:

1. (1)

The sequence $\{A_{n}\}_{n\in{\mathbb{Z}}_{+}}$ should satisfy the growth bound $\|A_{n}\|\leq M^{n}$ for some $M>0.$ 2. (2)

For $B$ to be positive (resp. self-adjoint), $A_{n}$’s should be positive (resp. self-adjoint). For $B$ to be isometric or unitary, $A_{n}$’s should be contractive.

The condition $(1)$ is natural as we are looking for dilations which are bounded operators. The condition $(2)$ is also obvious.

We now briefly describe the plan of the article. We recall and summarize some known answers to the first question of Problem 1.2 in Theorem 2.1, Theorem 2.4, Theorem 2.6, Theorem 3.1, and Theorem 4.1. To be more precise, in Theorem 2.1, we provide necessary and sufficient conditions of operator valued moment sequence that admit self-adjoint dilation. This result was initially obtained by Sz.-Nagy [26]. However, we provide a contemporary approach of dilation using the theory of completely positive maps. In Theorem 2.1, and Theorem 2.4, we have necessary and sufficient criteria for self-adjoint dilation problem in terms of Hankel matrices. This can be treated as the operator analog of the Hamburger moment problem. Finding necessary and sufficient conditions for an operator sequence to have positive dilation referred to the Hausdorff moment problem. In Theorem 2.6, we obtain necessary and sufficient conditions for operator valued moment sequences to have positive dilation. In Theorem 3.1, we have necessary and sufficient condition for an operator valued sequence to have unitary dilation. This can be treated as an operator analog of Toeplitz moment problem.

The main focus of this article is the third question. We try to obtain block operator forms for various classes of dilations. In Theorem 2.9, we show that if $\{A_{n}\}_{n\geq 0}$ admits self-adjoint dilation (say $B$) then $B$ has a tri-diagonal form (see Theorem 2.8) whose blocks are given by the recursive relation (see Section 2). In Theorem 3.2, we show that if $\{A_{n}\}_{n\geq 0}$ admits isometric dilation (say $V$), then $V$ has the form as in Equation (27) and blocks of $V$ are given by the recursive relation described in Section 3.

Then we focus on operator valued sequences of so called $\mathcal{C}_{A}$-class that admit isometric dilations. We present several necessary and sufficient conditions for an operator that belong to $\mathcal{C}_{A}$-class (see Theorem 4.1 and Theorem 4.2). In the final section, a special sub-class of $\mathcal{C}_{A}$-class operators for which we can write down isometric and unitary dilations explicitly in block operator form has been studied (see Theorem 5.3, and Theorem 5.5). Moreover, we describe their minimal dilation spaces (see Proposition 5.4 and Remark 5.6).

2. Self-adjoint and positive dilations

2.1. Classical moment problems

This is a well known topic in analysis. The subject has received considerable attention over the years, beginning with the pioneering efforts of Stieltjes, Riesz, Hamburger, Hausdorff and Kreĭn, followed by the works of Haviland, Akhiezer, Fuglede, Berg, Atzmon, and many others (see for example [4, 32] and references therein). For the convenience of the reader we recall a few basic results from this theory, which are relevant for our current discussion.

Given a sequence $\{m_{n}:n\geq 0\}$ of real numbers, the moment problem discusses the existence of a measure $\mu$ supported on a set $K\subseteq\mathbb{R}$ such that

[TABLE]

Determining the existence of such a measure $\mu$ is known as $K$-moment problem [33]. Three specific choices of $K$ stand out due to their natural importance and for historical reasons.

For a real sequence $\{m_{n}:\;n\geq 0\},$ the Hamburger moment problem is determining as when does there exist a positive Radon measure $\mu$ on $\mathbb{R}$ such that for all $n\geq 0,$ the integral $\int\limits_{-\infty}^{\infty}x^{n}\;d\mu(x)$ converges and satisfies

[TABLE]

Similarly, the well known Stieltjes moment problem and Hausdorff moment problem ask for existence of such measures $\mu$ (see Equation (4)) supported on $[0,\infty)$ and $[0,1]$ respectively. Equivalent conditions for the existence of solutions of these moment problems are very well known [4, 32] (also see references therein). In fact, for each $n\geq 0$, the associated Hankel matrices of the moment sequence are defined by

[TABLE]

It is very well known that the Hamburger moment problem has a solution if and only if the associated Hankel matrix

[TABLE]

Further, the Stieltjes moment problem has a solution if and only if the associated Hankel matrices

[TABLE]

Furthermore, the Hausdorff moment problem has a solution if and only if the sequence $\{m_{n}:\;n\geq 0\}$ is completely monotonic i.e.,

[TABLE]

where

[TABLE]

One may also consider measures supported on subsets of the complex plane. For instance see [3].

Coming to dilations of operator sequences, clearly the starting point is the following famous theorem of Sz.-Nagy. Let $T\in\mathcal{B(H)}$ and consider the operator valued sequence $\{T^{n}\}_{n\geq 0}$. This sequence admits a minimal isometric or unitary dilation if and only if $T$ is a contraction. Moreover, the minimal dilations are unique up to unitary equivalence. ( See [25] or [28, Theorem 1.1]). We call them as dilations (or power dilations) of $T$. In 1955, Schäffer [34] provided an explicit construction of minimal isometric and unitary dilations as follows: Let $D_{T}=(I-T^{*}T)^{\frac{1}{2}},D_{T^{*}}=(I-TT^{*})^{\frac{1}{2}}$ and $\mathcal{D}_{T}:=\overline{\rm range}D_{T},$ $\mathcal{D}_{T^{*}}=\overline{\rm range}D_{T^{*}}$. Take

[TABLE]

Then

[TABLE]

on $\mathcal{K}$ is a minimal isometric dilation of $T$. Take $\mathcal{L}=\cdots\oplus\mathcal{D}_{T^{*}}\oplus\mathcal{D}_{T^{*}}\oplus\mathcal{H}\oplus\mathcal{D}_{T}\oplus\mathcal{D}_{T}\oplus\cdots$ and define $U$ on $\mathcal{L}$ by

[TABLE]

The bold font indicates the location of operator $T$ from $\mathcal{H}$ to $\mathcal{H}$ and [math] entries are not displayed. In this article we provide Schäffer type constructions for several operator valued moment sequences.

2.2. Self-adjoint dilations

In 1952, Sz-Nagy obtained the following necessary and sufficient condition for a sequence of operators to admit self-adjoint dilation with spectrum contained in a given compact set (see Theorem 2.1). For reader’s convenience we present this result using the theory of completely positive maps. This method seems to have become a standard method to prove existence of operator dilations.

Theorem 2.1**.**

[26]** Let $X\subseteq\mathbb{R}$ be a compact set. Let $\{A_{n}:\;n\geq 0\}$ be a sequence of bounded self-adjoint operators on a Hilbert space $\mathcal{H}$ with $A_{0}=I$. It admits a self-adjoint operator dilation $B$ with $\sigma(B)\subseteq X$ if and only if

[TABLE]

whenever the complex polynomial $c_{0}+c_{1}x+\cdots+c_{n}x^{n}\geq 0\;$ for all $x\in X.$

Proof.

Suppose Equation (5) is satisfied. Then by the functional calculus,

[TABLE]

for every $g\in\mathcal{H}$ whenever $\sum\limits_{i=0}^{n}c_{i}x^{i}$ is positive on $X$. To prove the converse, let us define a map $\varphi\colon C(X)\to\mathcal{B}(\mathcal{H})$ by

[TABLE]

where $C(X)$ is the $C^{*}$-algebra of continuous functions on $X$. Let $\mathcal{P}(X)$ denote the algebra of all polynomials over $X$, then $\mathcal{P}(X)$ is a $\ast$-subalgebra of $C(X)$, it separates points of $X$ and hence $\mathcal{P}(X)$ is dense in $C(X)$ by Stone-Weierstrass theorem. From the hypothesis, it follows that $\varphi(p)\geq 0$ whenever $p\in\mathcal{P}(X)$ is positive. So $\varphi$ can be extended to a positive map on $C(X)$. Thus $\varphi$ is completely positive as $C(X)$ is a commutative $C^{\ast}$-algebra. Therefore by Stinespring’s theorem, there is a Hilbert space $\mathcal{K}$, an isometry $V\colon\mathcal{H}\to\mathcal{K}$ and a unital $\ast$-homomorphism $\pi\colon C(X)\to\mathcal{B}(\mathcal{K})$ such that

[TABLE]

Let $\pi(x)=B$. Then $B^{\ast}=\pi(x)^{\ast}=\pi(x^{\ast})=\pi(x)=B$ since $X\subset\mathbb{R}$. This implies that

[TABLE]

Identifying $h\in\mathcal{H}$ with $Vh\in V(\mathcal{H})$ the proof is complete. ∎

Remark 2.2**.**

It follows from the proof of Theorem 2.1 that if a sequence $\{A_{n}:\;n\geq 0\}$ admits self-adjoint dilation then

[TABLE]

since $V$ is an isometry.

The existence of self-adjoint dilation for an operator sequence is also linked with positivity of the corresponding Hankel matrix. To see this we begin with the following observation.

Lemma 2.3**.**

Let $\{A_{n}\}_{n\geq 0}$ be a sequence in $\mathcal{B}(\mathcal{H})$ with $A_{0}=I$. Then $\sum\limits_{i,j=0}^{n}X_{i}^{\ast}A_{i+j}X_{j}\geq 0$ for any $X_{0},X_{1},\cdots,X_{n}\in\mathcal{B}(\mathcal{H}),n\geq 0$ if and only if the associated Hankel matrix

[TABLE]

Proof.

Suppose that $H_{n}\geq 0$ for all $n\geq 0$ and let $X_{0},X_{1},\cdots,X_{n}\in\mathcal{B}(\mathcal{H})$. For every $g\in\mathcal{H}$, we see that

[TABLE]

where $\widetilde{g}=\begin{bmatrix}X_{0}g\\ \vdots\\ X_{n}g\end{bmatrix}$. To prove the converse, choose and fix $g\in\mathcal{H}$ with $\|g\|=1.$ Now for $h_{0},h_{1},\cdots,h_{n}\in\mathcal{H}$, take $X_{i}=|h_{i}\rangle\langle g|\;(0\leq i\leq n)$. Then

[TABLE]

∎

Let $\{A_{n}\}_{n\geq 0}$ be a sequence of self-adjoint operators acting on a Hilbert space $\mathcal{H}$. Consider the associated Hankel matrices:

[TABLE]

In the following theorem we use both the method of completely positive maps as well as that of positive kernels.

Theorem 2.4**.**

Let $\{A_{n}\}_{n\geq 0}$ be a sequence of self-adjoint operators with $A_{0}=I$ and $\|A_{n}\|\leq 1$ for all $n$. Then it admits a self-adjoint contraction dilation if and only if $H_{n}\geq 0$ and $H^{(2)}_{n}\leq H_{n}$ for each $n.$

Proof.

Suppose there exists a self-adjoint contraction dilation $B$ on a Hilbert space $\mathcal{K}\supseteq\mathcal{H}$. Then by the proof of Theorem 2.1, there exists a completely positive map $\varphi:C[-1,1]\to\mathcal{B(H)}$ such that $\varphi(x^{m})=A_{m}$ for each $m\geq 0.$ Consider the element $L\in M_{n+1}(C[-1,1])$,where

[TABLE]

Then $L^{*}L\geq 0$ in $M_{n+1}(C[-1,1])$. Since $H_{n}=\varphi_{n+1}(L^{*}L),$ and $\varphi$ is completely positive, $H_{n}\geq 0.$ We define $G\in M_{n+1}(C[-1,1])$ by

[TABLE]

Then $G^{*}G\geq 0$ in $M_{n+1}(C[-1,1])$. Since $H_{n}-H^{(2)}_{n}=\varphi_{n+1}(G^{*}G),$ and $\varphi$ is completely positive, $H_{n}-H^{(2)}_{n}\geq 0.$

Conversely, assume that $H_{n}\geq 0$ and $H^{(2)}_{n}\leq H_{n}$ for each $n.$ Let $M:=\mathbb{Z}_{+}\times\mathcal{H}$. Define a map $k\colon M\times M\to\mathbb{C}$ by

[TABLE]

Since $H_{n}\geq 0$ for all $n\geq 0,$ $k$ is a positive definite kernel. Let $V$ be a vector space of all complex functions on $M$ which is zero except for finitely many points of $M$. Since $k$ is positive definite, $V$ is a semi-inner product space with respect to:

[TABLE]

Let us take $\mathcal{N}=\{\xi\in V:\;\langle\xi,\xi\rangle=0\}$. Then by Cauchy-Schwarz inequality, $\mathcal{N}=\{\xi:\;\langle\xi,\eta\rangle=0,\;\text{for every}\;\eta\in V\}$ is subspace of $V$. Take $\mathcal{K}$ as the Hilbert space obtained by the completion of the quotient space $V/\mathcal{N}$. Define $\lambda\colon M\to\mathcal{K}$ by

[TABLE]

Since $\langle\lambda(0,g),\lambda(0,h)\rangle=\langle g,h\rangle$ for every $g,h\in\mathcal{H}$, we see that $\mathcal{H}$ can be identified as a subspace of $\mathcal{K}$ via the map $g\mapsto\lambda(0,g).$ Moreover, $\mathcal{K}=\overline{\rm span}\{\lambda(m,g):\;m\geq 0,\;g\in\mathcal{H}\}$. Define $B(\lambda(m,g))=\lambda(m+1,g)$ for every $(m,g)\in M$. Then

[TABLE]

This implies that $B$ extends to a linear contraction. It is easy to see that it is self-adjoint. Moreover, for every $g,h\in\mathcal{H}$,

[TABLE]

Hence $B$ is a self-adjoint dilation of $\{A_{n}\}_{n\geq 0}.$ ∎

If a sequence $\{A_{n}:n\geq 0\}$ with $A_{0}=I$ admits a self-adjoint dilation and suppose $A_{2}=A_{1}^{2}$. Then from the characterizations above it is possible to see that $A_{n}=A_{1}^{n}$ for every $n\geq 1$. In fact, much stronger results are known now (see [27] for more details).

2.3. Positive dilations

Here it is convenient to have the following standard definition.

Definition 2.5**.**

A sequence $\{A_{n}\}_{n\geq 0}$ of bounded operators on a Hilbert space $\mathcal{H}$ is called completely monotone if $(-1)^{k}(\Delta^{k}A)_{n}\geq 0\text{ for every }n,k\geq 0,$ where

[TABLE]

In the classical setup, the Hausdorff moment problem has a solution if and only if the given sequence is completely monotone. The same result holds true for operator sequences also. We prove the result here by defining an appropriate positive definite kernel.

Theorem 2.6**.**

Let $\{A_{n}\}_{n\geq 0}$ be a sequence of positive operators in $\mathcal{B}(\mathcal{H})$ with $A_{0}=I$. Then it is completely monotone if and only if it admits a positive contraction dilation.

Proof.

Though the existence of such positive contraction is clear from the Theorem 2.1 when $X=[0,1]$, we provide an explicit proof for the construction of $B$. Suppose that $\{A_{n}:\;n\geq 0\}$ is completely monotone and let $M:=\mathbb{Z}_{+}\times\mathcal{H}$. Define the map $k\colon M\times M\to\mathbb{C}$ by

[TABLE]

Now, we show that $k$ is a positive definite kernel. Equivalently, it is enough to show that

[TABLE]

Let us define the map $\Phi(x^{n})=A_{n}$ for every $n$. By mathematical induction it can be shown that every polynomial $q(x)$ of degree $n$ with real coefficients over $[0,1]$ can be written using Bernstein polynomial (see [4, Page 76]) as follows:

[TABLE]

where $u_{j}(x)$ is polynomial of degree less than or equal to $n$ and is independent of $N.$ Since the sequence is completely monotone, for every $k,n\geq 0$, we see that

[TABLE]

If $q(x)$ is a positive polynomial over $[0,1]$, it follows from Equation (8) that

[TABLE]

This shows that $\Phi$ is a positive map. Since $\pm q\leq\|q\|_{\infty},$ it implies that $\Phi(q)\leq\|q\|_{\infty}\Phi(1)$ and thus $\Phi$ is continuous on the space of polynomials over $[0,1].$ Therefore $\Phi$ is a positive map on $C([0,1])$ and so it is completely positive map. For every $n\geq 0$, we have $H_{n}=\Phi_{n+1}(L^{\ast}L)\geq 0,$ where

[TABLE]

Hence $k$ is a positive definite kernel. Furthermore, if we denote (for $n\geq 0$)

[TABLE]

then

[TABLE]

where

[TABLE]

Let $V$ be a vector space of all complex functions on $M$ which is zero except for finitely many points of $M$. Then $V$ is a semi-inner product space with respect to:

[TABLE]

Let us take $\mathcal{N}=\{\xi\in V:\;\langle\xi,\xi\rangle=0\}$. Thus by Cauchy-Schwarz inequality, it follows that $\mathcal{N}=\{\xi:\;\langle\xi,\eta\rangle=0,\;\text{for every}\;\eta\in V\}$. Take $\mathcal{K}$ as the Hilbert space obtained by the completion of the quotient space $V/\mathcal{N}$. Define $\lambda\colon M\to\mathcal{K}$ by

[TABLE]

Since $\langle\lambda(0,g),\lambda(0,g)\rangle=\|g\|^{2}$ for every $g\in\mathcal{H}$, we see that $\mathcal{H}$ can be identified as a subspace of $\mathcal{K}$ via the map $g\mapsto\lambda(0,g).$ Moreover, $\mathcal{K}=\overline{\rm span}\{\lambda(m,g):\;m\geq 0,\;g\in\mathcal{H}\}$. Define $B(\lambda(m,g))=\lambda(m+1,g)$ for every $(m,g)\in M$. Then

[TABLE]

and

[TABLE]

This implies that $B$ defines a contractive and positive operator. Finally, for every $g,h\in\mathcal{H}$, we see that

[TABLE]

Therefore, every monotone sequence admits a contractive positive dilation. Conversely, suppose that there is a positive contraction $B$ satisfying, A_{n}=P_{\mathcal{H}}B^{n}\big{|}_{\mathcal{H}} for $n\geq 0$ then

[TABLE]

for every $k,n\geq 0.$ Therefore, $\{A_{n}\}_{n\geq 0}$ is a completely monotone sequence. ∎

2.4. Concrete self-adjoint dilations

Now we turn our discussion to a concrete construction of self-adjoint dilation. Before that, let us recall some known facts from the literature that a bounded self-adjoint operator say $B$ defined on a separable Hilbert space with a unit cyclic vector $v$ can be represented by a tridiagonal matrix with respect to the basis obtained by Gram-Schmidt process from the set $\{v,Bv,B^{2}v,\cdots\}$. Suppose the matrix $B$ is expressed as,

[TABLE]

then it is related to a family of monic orthogonal polynomials given by $p_{0}(x)=1,p_{1}(x)=(x-a_{0})$,

[TABLE]

Note that here equation (12) is obtained by expanding the determinant of $(xI-B_{n})$ using the last row, where $B_{n}$ is the $n\times n$ truncated matrix of $B$. More information on this can be found in the well known classic on Hilbert space linear transformations [37]. Some applications of this idea to quantum theory with worked out examples can be seen in [12].

Let $\mu(\cdot)=\|E(\cdot)v\|^{2}$ be the probability measure defined on the Borel $\sigma-$field of the real line, where $E$ is the spectral measure associated to $B$. Clearly it is supported on the spectrum of $B$. Then $\{p_{n}(x):\;n\geq 0\}$ is an orthonormal basis of $L^{2}(\mu)$ (for details, see [37]). Moreover, $m$-th moment of $\mu$ is given by

[TABLE]

Conversely, given any compactly supported probability measure $\mu$ on the real line we can take $B$ as the operator ‘multiplication by $x$’, on $L^{2}(\mu)$ and cyclic vector $v$ as the constant function $1$. We can observe the tridiagonal form of $B$ on the basis normalized orthogonal polynomials. The coefficients $\{a_{n},b_{n}:n\geq 0\}$ are known as Jacobi parameters of the measure $\mu.$ Here $B$ is a self-adjoint dilation of the moment sequence of the probability measure. This motivates us to construct such tridiagonal operator matrix $B$ for a self-adjoint dilation of an operator sequence $\{A_{n}:\;n\geq 0\}.$ Such tri-diagonal blocks, known as generalized Jacobi 3-diagonal relations are well known in quantum theory (See [1]).

Lemma 2.7**.**

Let $B$ be a bounded operator on some Hilbert space ${\mathcal{K}}$. Let ${\mathcal{H}}$ be a closed subspace of ${\mathcal{K}}.$ Assume

[TABLE]

Then ${\mathcal{K}}$ decomposes as a direct sum of Hilbert spaces

[TABLE]

where ${\mathcal{H}}_{0}={\mathcal{H}}$ and with respect to this decomposition the operator $B$ has the ‘upper Hessenberg’ form:

[TABLE]

where $\mathcal{H}_{n}=\overline{B_{n(n-1)}(\mathcal{H}_{n-1})}$ for every $n$. Conversely any such $B$ satisfies Equation (13).

Proof.

Take ${\mathcal{H}}_{n]}=\overline{\rm span}\{B^{m}h:0\leq m\leq n-1,h\in{\mathcal{H}}\}$ and ${\mathcal{H}}_{n}={\mathcal{H}}_{n]}\bigcap{{\mathcal{H}}_{(n-1)]}^{\perp}}$ for $n\geq 2$ with ${\mathcal{H}}_{1]}={\mathcal{H}}_{1}.$ Then clearly ${\mathcal{K}}=\oplus_{n\geq 1}{\mathcal{H}}_{n},$ and as $B({\mathcal{H}}_{n]})\subseteq{\mathcal{H}}_{(n+1)]}$, $B({\mathcal{H}}_{n})\subseteq\oplus_{m=0}^{n+1}{\mathcal{H}}_{m+1}$. Consequently, the operator $B$ has the form described above. The range and condition and the converse statements are easy to see. ∎

Theorem 2.8**.**

Let $\{A_{n}\}_{n\geq 0}$ be a sequence of self-adjoint operators in $\mathcal{B}(\mathcal{H})$ with $A_{0}=I$, admitting a minimal self-adjoint dilation $B$ in $\mathcal{B}(\mathcal{K})$ for some Hilbert space $\mathcal{K}$. Then the space $\mathcal{K}=\mathcal{H}_{0}\oplus\mathcal{H}_{1}\oplus\cdots$ ( $\mathcal{H}_{0}=\mathcal{H})$, so that the operator $B$ has the tridiagonal form:

[TABLE]

and $\mathcal{H}_{n}=\overline{B_{n(n-1)}(\mathcal{H}_{n-1})}$ for all $n\geq 1.$

Proof.

From the previous lemma, $B$ has upper Hessenberg form. But since $B$ is self-adjoint, $B_{ij}=B_{ji}^{*}=0$ for $j>(i+1).$

∎

Consider the set up of this theorem. We try to determine the blocks using the sequence $\{A_{n}\}_{n\geq 0}$ and the dilation property. It is done recursively and to do this we need to invert some operators. In general, it is quite possible that some of these operators are not invertible and we may have to modify the construction. For the moment we assume invertibility of concerned operators as and when required.

Since $B$ is a self-adjoint dilation for the moment sequence $\{A_{n}\}_{n\geq 0}$ we have $(B^{n})_{00}=A_{n}$ for every $n\geq 0$. Clearly, $B_{00}=A_{1}$ and $B_{10}$ is obtained as,

[TABLE]

That is, $B_{10}^{\ast}B_{10}=A_{2}-A_{1}^{2}$. Since $(A_{2}-A_{1}^{2})\geq 0$ (see Remark 2.2), we can choose $B_{10}=(A_{2}-A_{1}^{2})^{1/2}.$ Firstly, we explain the process to compute the diagonal block $B_{11}$ and then establish the recurrence relation to obtain diagonal and lower diagonal blocks. From the Equation (14), we have

[TABLE]

since $B_{0r_{1}}B_{r_{1}r_{2}}B_{r_{2}0}=0$ when either $r_{1}$ or $r_{2}$ is 2. This implies that

[TABLE]

By substituting the first column information we get

[TABLE]

The formulae for diagonal and off-diagonal blocks are as follows. Firstly, note that each diagonal block is a self-adjoint operator and it is computed by the compression of odd powers of $B$. Suppose that $(n-1)$ columns of $B$ are known then $B_{nn}$ is computed as follows:

[TABLE]

This implies that

[TABLE]

Now notice that the lower diagonal block in the first column is given by $B_{21}=(A_{2}-A_{1}^{2})^{1/2}$. These lower diagonal blocks can be obtained by the compression of even powers of $B.$ Suppose that $(n-1)$ columns of $B$ are known then $B_{(n+1)n}$ is obtained as below:

[TABLE]

This implies that

[TABLE]

In this case, we can choose

[TABLE]

With these computations and Lemma 2.7, we get the following result.

Theorem 2.9**.**

Let $\{A_{n}\}_{n\geq 0}$ be a sequence of self-adjoint operators in $\mathcal{B}(\mathcal{H})$ with $A_{0}=I$, admitting a self-adjoint dilation. If the inverses appearing in the recurrence relations above are well-defined bounded operators, then these formulae provide a self-adjoint minimal dilation with block tridiagonal form as above.

Example 2.10**.**

Let $T\in\mathcal{B(H)}$ be a self-adjoint operator. We define a moment sequence $\{A_{n}\}_{n\geq 0}$ by

[TABLE]

Then $V$ acting on $\mathcal{H}\oplus\mathcal{H}\oplus\mathcal{H}$,

[TABLE]

is a self-adjoint dilation of $\{A_{n}:n\geq 0\}.$

This example can be generalized as follows. Suppose $\{m_{n}:n\geq 0\}$ is a moment sequence of a compactly supported probability measure $\mu$ on $\mathbb{R}.$ Let $T$ be a bounded self-adjoint operator on a Hilbert space $\mathcal{H}$. Then $\{m_{n}T^{n}:n\geq 0\}$ is a operator moment sequence admitting self-adjoint dilation. Take the dilation space $\mathcal{K}=\mathcal{H}\otimes L^{2}(\mu)$ with $\mathcal{H}$ identified as a subspace by identifying $h\in\mathcal{H}$ with $h\otimes 1$, where $1$ is the constant function 1 in $L^{2}(\mu).$ Let $B$ be the tri-diagonal form of multiplication by ‘$x$’ operator on $L^{2}(\mu)$ with cyclic vector $1$ as in Equation (11). Then $T\otimes B$ is a self-adjoint dilation of $\{m_{n}T^{n}:n\geq 0\}$:

[TABLE]

An operator $T\in\mathcal{B(H)}$ is called quasinormal if $T(T^{*}T)=(T^{*}T)T.$ We recall that $T$ is quasinormal if and only if $(T^{*}T)^{k}=T^{*k}T^{k}$ for all $k\in\mathbb{Z}_{+}$ [27].

Example 2.11**.**

Let $T\in\mathcal{B(H)}$ be a quasinormal operator. Then the moment sequence $\{A_{n}\}_{n\geq 0}$ defined by

[TABLE]

admits a self-adjoint dilation. In fact, $V$ acting on $\mathcal{H}\oplus\mathcal{H}\oplus\mathcal{H}$ defined by

[TABLE]

is a self-adjoint dilation of $\{A_{n}:n\geq 0\}.$

Like before, this example also can be generalized to have self-adjoint dilations for operator moment sequences $\{1,0,m_{2}T^{*}T,0,m_{4}(T^{*})^{2}T^{2},0,\ldots\}$, where $\{m_{n}:n\geq 0\}$ is moment sequence of a compactly supported probability measure $\mu$ on $\mathbb{R}$, which is symmetric around [math]. Such a symmetry ensures that odd moments and also diagonal Jacobi parameters of the measure are all equal to 0.

3. Unitary and isometric dilations

In this section, we mainly discuss necessary and sufficient conditions on operator sequences to admit unitary or isometric dilations. Let $\{A_{n}\}_{n\geq 0}$ be a sequence in $\mathcal{B}(\mathcal{H})$ with $A_{0}=I$. Then $\{A_{n}\}_{n\geq 0}$ is said to admit a unitary (respectively isometric) dilation if there is a Hilbert space $\mathcal{K}$ containing $\mathcal{H}$ and a unitary (respectively unitary) $U\in\mathcal{B}(\mathcal{K})$ such that

[TABLE]

Obviously every unitary dilation is an isometric dilation. Sz.-Nagy dilation theorem implies in particular that every isometry has a power dilation to a unitary. Therefore, if a sequence admits an isometric dilation it also admits a unitary dilation. Consequently, an operator sequence admits an isometric dilation if and only if it admits a unitary dilation. However, there is a difference in the notion of minimality. With notation as above, a unitary dilation $U$ acting on $\mathcal{K}$ is minimal if

[TABLE]

On the other hand, an isometric dilation $U$ acting on $\mathcal{K}$ is minimal if

[TABLE]

The classical Szegő kernel and Poisson kernel [20] (also see [15] for more details on Poisson kernel) are denoted and defined by

[TABLE]

A Poisson kernel can be written using the Szegő kernel:

[TABLE]

Moreover, $P_{r}(\theta)\geq 0$ for all $0\leq r<1$ and $0\leq\theta\leq 2\pi.$

Keeping Szegő and Poisson kernels in mind, we define an operator valued kernel function for a sequence of operators. Let ${\bf A}=\{A_{n}\}_{n\geq 0}$ be a sequence of contractions in $\mathcal{B(H)}$. The associated Szegő kernel function $S_{\bf A}:\mathbb{D}\to\mathcal{B(H)}$ is defined as

[TABLE]

We define the associated Poisson kernel function by

[TABLE]

F. H. Vasilescu introduced operator valued Poisson kernel functions [40] for a $d$-tuples of operators using defect operators to study Holomorphic functional calculus. Such Poisson kernel functions can not be extended to operator valued sequences as we do not have semigroup property (i.e. $A_{n}A_{m}=A_{n+m}$ may not hold). In the next result, we discuss necessary and sufficient criteria for isometric and unitary dilations in terms of the Poisson kernel.

Theorem 3.1**.**

Let $\{A_{n}\}_{n\geq 0}$ be a sequence in $\mathcal{B}(\mathcal{H})$ with $A_{0}=I$. Then the following are equivalent:

(i) $\{A_{n}\}_{n\geq 0}$ admits a unitary/isometric dilation.

(ii)

[TABLE]

where $A_{-n}:=A^{*}_{n}$ for $n\in{\mathbb{N}}.$

(iii) $P_{\bf A}(z)\geq 0$ in $\mathcal{{B(H)}}$ for all $z\in\mathbb{D}.$

Before coming to the proof of this result, we point out that there is a subtle point here. The condition in Equation (18) is a priori weaker than having complete positivity of the $B({\mathcal{H}})$ valued kernel $K:{\mathbb{Z}}\times{\mathbb{Z}}\to B({\mathcal{H}})$ defined by:

[TABLE]

as that would mean,

[TABLE]

Proof.

$(i)\implies(ii)$ Suppose assume that $(i)$ holds true. Then the Equation(15) holds true. For every $N\geq 0$ and $g\in\mathcal{H}$, we see that

[TABLE]

$(ii)\implies(i)$ Consider the operator system \mathcal{S}=\Big{\{}\sum\limits_{k=-N}^{N}c_{k}e^{ik\theta}:\;N\geq 0\Big{\}}\subset C(\mathbb{T}). Define $\Phi\colon\mathcal{S}\to\mathcal{B}(\mathcal{H})$ by $\Phi(e^{in\theta})=A_{n}$ for every $n\in\mathbb{Z}$. Suppose that $f\in\mathcal{S}$ is strictly positive then by Fejér-Riesz theorem (see [28, Lemma 2.5]), we see that $f(e^{i\theta})=\sum\limits_{i,j=0}^{N}\overline{c_{\ell}}c_{k}e^{i(k-\ell)\theta}$ for some $c_{0},c_{1},\ldots,c_{N}$ and hence

[TABLE]

Further, if $g\in\mathcal{S}$ is non-negative, then $g+\epsilon\cdot 1$ is strictly positive for every $\epsilon>0$ and so $\Phi(g+\epsilon\cdot 1)=\Phi(g)+\epsilon I\geq 0$ for every $\epsilon>0$. Thus $\Phi(g)\geq 0,$ it follows that $\Phi$ is a positive map and hence it is bounded. By Stone-Weierstrass theorem, $\mathcal{S}$ is dense in $C(\mathbb{T})$. So $\Phi$ can be extended to a positive map on $C(\mathbb{T})$ by [28, Exercise 2.2]. Again we denote it by $\Phi$. Therefore, it follows from [28, Theorem 3.11] that $\Phi$ is completely positive map on $C(\mathbb{T})$.

Now by Stinespring’s theorem there exists a triple $({\mathcal{K}},\pi,V)$ where ${\mathcal{K}}$ is a Hilbert space, $\pi:C({\mathbb{T}})\to B({\mathcal{K}})$ is a unital $*$-homomorphism, $V\colon{\mathcal{H}}\to{\mathcal{K}}$ a bounded linear map satisfying

[TABLE]

Since $\Phi$ is unitary, $V$ is an isometry and we may consider ${\mathcal{H}}$ as a subspace of ${\mathcal{K}}$ by identifying $h\in{\mathcal{H}}$ with $Vh$ in ${\mathcal{K}}$. Then the Equation (20) reads as

[TABLE]

Since the function $f_{0}(z)=z$ is a unitary, $\pi(f_{0})$ is a unitary in ${\mathcal{B}(K)}$. Taking $U=\pi(f_{0})$ and considering $f_{0}^{n}(z)=z^{n},$ for $n\geq 0$, we get

[TABLE]

$(i)\implies(iii)$ Suppose $\{A_{n}:n\geq 0\}$ admits a unitary dilation. Then there is a unitary operator $U\in\mathcal{B(K)}$ for some Hilbert space $\mathcal{K}\supseteq\mathcal{H}$ such that

[TABLE]

First notice that for each $z\in\mathbb{D},$ $(1-zU^{*})$ is invertible and $(1-zU^{*})^{-1}=1+zU^{*}+z^{2}U^{*2}+\cdots$ and

[TABLE]

Now a simple computation ensures that $(I-zU^{*})^{-1}+(I-\overline{z}U)^{-1}-I=(1-|z|^{2})(I-\overline{z}U)(I-\overline{z}U)^{*}.$ Also, it is immediate to see that $(I-zU^{*})^{-1}+(I-\overline{z}U)^{-1}-I\geq 0$ as $|z|<1.$ It implies that P_{\mathcal{H}}\Big{(}(I-zU^{*})^{-1}+(I-\overline{z}U)^{-1}-I\Big{)}|_{\mathcal{H}}\geq 0. Using the hypothesis of dilation Equation (21), we see that

[TABLE]

Thus the series $\sum_{n=0}^{\infty}z^{n}A^{*}_{n}$ converges in norm absolutely for all $z\in\mathbb{D}.$ Therefore the map $S_{\bf A}(z):\mathbb{D}\to\mathcal{B(H)}$ is well defined. Further, we compute that

[TABLE]

Since P_{\mathcal{H}}\Big{(}(I-zU^{*})^{-1}+(I-\overline{z}U)^{-1}-I\Big{)}|_{\mathcal{H}}\geq 0, therefore we have $P_{\bf A}(z)\geq 0$ for all $z\in\mathbb{D}.$

$(iii)\implies(i)$ Assume that $P_{\bf A}(z)\geq 0$ for all $z\in\mathbb{D}.$ Let $0\leq r<1,$ then for each $l\in\mathbb{Z},$ we have

[TABLE]

Let us define a map $\varphi$ from an operator system $\mathcal{S}$ to $\mathcal{B(H)}$ by

[TABLE]

for all $\sum_{n=0}^{N}p_{n}e^{in\theta}+\sum_{m=0}^{N}\overline{q_{m}}e^{-im\theta}\in\mathcal{S}.$ Now it is enough to show that the map $\varphi:\mathcal{S}\to\mathcal{B}(\mathcal{H})$ is a positive map. Now, we observe the following

[TABLE]

Similarly, we have

[TABLE]

Combining all, we finally obtain that

[TABLE]

Notice that for all $0\leq\theta\leq 2\pi$ and $0\leq r<1,$ we have $P_{\bf A}(re^{i\theta})\geq 0.$ It follows that \int_{0}^{2\pi}\big{(}\sum_{n=0}^{N}p_{n}e^{in\theta}+\sum_{m=0}^{N}\overline{q_{m}}e^{-im\theta}\big{)}P_{\bf A}(re^{i\theta})d\theta\geq 0 whenever $\sum_{n=0}^{N}p_{n}e^{in\theta}+\sum_{m=0}^{N}\overline{q_{m}}e^{-im\theta}\geq 0.$ These completes the proof as $\varphi$ is a positive map. ∎

Let $V$ be minimal isometric dilation on some Hilbert space ${\mathcal{K}}$ for a sequence of contractions $\{A_{n}\}_{n\geq 0}$ on $\mathcal{H}.$ In particular, ${\mathcal{K}}=\overline{\rm span}\{V^{n}\mathcal{H}:n\geq 0\}$. Then by Lemma 2.7, ${\mathcal{K}}$ decomposes as

[TABLE]

where ${\mathcal{H}}_{0}={\mathcal{H}}$ and with respect to this decomposition the operator $V$ has the block form:

[TABLE]

We wish to construct $V_{ij}$’s using the given sequence $\{A_{n}\}.$ Recall that ${\mathcal{H}}_{n]}=\overline{\mbox{s}pan}\{V^{m}h:0\leq m\leq n-1,h\in{\mathcal{H}}\}$ and ${\mathcal{H}}_{n}={\mathcal{H}}_{n]}\bigcap{{\mathcal{H}}_{(n-1)]}^{\perp}}$ for $n\geq 1$ with ${\mathcal{H}}_{0]}={\mathcal{H}}_{0}.$

We try to determine the blocks using the sequence $\{A_{n}\}_{n\geq 0}$ and the dilation property. Like in the case of self-adjoint dilation we assume invertibility of concerned operators as and when required. Since $V$ is an isometric dilation for the moment sequence $\{A_{n}\}_{n\geq 0}$ we have P_{\mathcal{H}}V^{n}\big{|}_{\mathcal{H}}=A_{n}. In other words, $(V^{n})_{00}=A_{n},\;\text{for}\;n\geq 0.$ Let us compute the first column of $V$. Clearly, $V_{00}=A_{1}$ and since $V$ is an isometry, we get

[TABLE]

One can choose that $V_{10}=(I-A_{1}^{\ast}A_{1})^{1/2}$ as $A_{1}$ is a contraction. The first row of $V$ is given by the following recursive relation: For every $n\in\mathbb{N},$ we have

[TABLE]

Since $V_{ij}=0$ whenever $i>j+1$, it implies that

[TABLE]

Therefore,

[TABLE]

Next, we use the isometric property of $V$ to obtain expression for the $n^{\text{th}}$ column of $V$. For a fixed $n\in\mathbb{N}$, suppose the first $(n-1)$ columns are known then $V_{1n}$ is given by considering the inner product of the first column with the $n^{\text{th}}$ column which is zero. That is, $V_{00}^{\ast}V_{0n}+V_{10}^{\ast}V_{1n}=0,$ and therefore

[TABLE]

Indeed $V_{kn}$ for $0<k\leq n$ is given by the inner product of the $(k-1)^{\text{th}}$ column with the $n^{\text{th}}$ column which is again zero. That is, $\sum\limits_{i=0}^{k}V^{\ast}_{i(k-1)}V_{in}=0.$ Equivalently, $V^{\ast}_{k(k-1)}V_{kn}+\sum\limits_{i=0}^{k-1}V^{\ast}_{i(k-1)}V_{in}=0.$ This implies that

[TABLE]

For $k=n+1$, we have $\sum\limits_{i=0}^{n+1}V_{in}^{\ast}V_{in}=I$ and equivalently, $V_{(n+1)n}^{\ast}V_{(n+1)n}+\sum\limits_{i=0}^{n}V_{in}^{\ast}V_{in}=I.$ It implies that

[TABLE]

One can choose that

[TABLE]

As a result, the recurrence relations are given by $V_{00}=A_{1},\;V_{10}=(I-A_{1}^{\ast}A_{1})^{1/2}$,

[TABLE]

and

[TABLE]

for $n>0.$

The computations above lead to the following result.

Theorem 3.2**.**

Let $\{A_{n}\}_{n\geq 0}$ be a sequence of contractions on $\mathcal{H}$ and $A_{0}=I$ admitting an isometric dilation. If the inverses appearing in the recurrence relations above are well defined bounded operators, then these formulae provide a minimal isometric dilation with the blocks of $V$ described as above.

Now we look at unitary dilations.

Theorem 3.3**.**

Suppose $U$ is a unitary on a Hilbert space $\mathcal{K}$ and $\mathcal{H}$ is a closed subspace of $\mathcal{K}$, satisfying $\mathcal{K}=\overline{\rm span}\{U^{n}h:h\in\mathcal{H},n\in\mathbb{Z}\}.$ Then ${\mathcal{K}}$ decomposes as

[TABLE]

where ${\mathcal{H}}_{0}={\mathcal{H}}$ so that with respect to this decomposition $U$ has the form:

[TABLE]

Proof.

Take ${\mathcal{H}}_{0]}={\mathcal{H}}_{0}$ and ${\mathcal{H}}_{n]}=\overline{\mbox{span}}\{U^{m}h:0\leq m\leq n-1,h\in{\mathcal{H}}\}$ and ${\mathcal{H}}_{n}={\mathcal{H}}_{n]}\bigcap{{\mathcal{H}}_{(n-1)]}^{\perp}}$ for $n\geq 1.$ $\mathcal{K}_{+}=\bigvee^{\infty}\limits_{n=0}U^{n}\mathcal{H},$ and $\mathcal{K}_{-}=\mathcal{K}^{\perp}_{+}.$ Then clearly ${\mathcal{K}}_{+}=\oplus_{n\geq 0}{\mathcal{H}}_{n}$ and $U({\mathcal{H}}_{n]})\subseteq{\mathcal{H}}_{(n+1)]}.$ It implies that $n$-th column of the block matrix satisfies the desired property for all $n\geq 0.$ Notice that $U^{*}(\mathcal{K}_{-})\subseteq\mathcal{K}_{-}$ as $\mathcal{K}_{-}=\mathcal{K}^{\perp}_{+}$ and $U(\mathcal{K}_{+})\subseteq\mathcal{K}_{+}.$ We claim that

[TABLE]

Let $x\in\bigcap\limits_{n=0}^{\infty}U^{*n}(\mathcal{K}_{-})=\{0\}.$ Let $n\geq 0,$ and $h\in\mathcal{H}$ notice that $x=U^{*n}x_{n}$ for some $x_{n}\in\mathcal{K}_{-}.$ Then $\langle x,U^{*n}h\rangle=\langle U^{*n}x_{n},U^{*n}h\rangle=\langle x_{n},h\rangle=0,$ and $\langle x,U^{m}h\rangle=0$ for all $m\geq 0.$ It follows that $x=0.$ Hence $U^{*}:\mathcal{K}_{-}\to\mathcal{K}_{-}$ is a shift with wandering subspace

[TABLE]

Therefore $\mathcal{K}_{-}$ can be decomposed as

[TABLE]

Now, it is enough to see that $U(\mathcal{H}_{-1})\subseteq\mathcal{K}_{+}.$ Consider $x\in\mathcal{H}_{-1},$ then for any $y\in\mathcal{K}_{-},$ we see that $\langle Ux,y\rangle=\langle x,U^{*}y\rangle=0,$ as $x\perp U^{*}\mathcal{K}_{-}.$ This completes the proof. ∎

4. $\mathcal{C}_{A}$-class operators

Inspired by Sz.-Nagy’s dilation theorem for contractions, Berger and Stamfli[11], obtained the following interesting theorem. Suppose $T\in\mathcal{B(H)}$. Then the numerical radius $w(T)\leq 1$ if and only if the operator sequence $\{\frac{1}{2}T^{n},\;n\geq 1\}$ admits a unitary dilation. This led to the following definition. Fix $\rho>0$. Then a bounded operator $T\in\mathcal{B(H)}$ is said to be in $\mathcal{C}_{\rho}$-class if there is a unitary operator $U$ on some Hilbert space $\mathcal{K}\supset\mathcal{H}$ such that T^{n}=\rho P_{\mathcal{H}}U^{n}\big{|}_{\mathcal{H}} for $n\geq 1$ (see page 43 of [25]). Thanks to Sz.-Nagy’s dilation theorem, $\mathcal{C}_{1}$-class is precisely the set of all contractions in $\mathcal{B(H)}$. Similarly from the Berger-Stamfli’s Theorem [11] we know that $\mathcal{C}_{2}$-class is the set of all operators in $\mathcal{B(H)}$ whose numerical radius is less than equal to one. An operator $T$ is in $\mathcal{C}_{\rho}$-class (see Theorem 11.1 of [25]) if and only if

[TABLE]

M. A. Dritschel, H. J. Woerdeman [23] have developed a model theory for the $\mathcal{C}_{2}$-class. Over the past few decades there has been an extensive study of $\mathcal{C}_{\rho}$-class operators. To mention some of them we refer [5, 10, 11, 22, 17, 18, 24] and references therein. Generalizing this notion in a natural way H. Langer introduced the $\mathcal{C}_{A}$-class. Let $A\in\mathcal{B(H)}$ be a positive invertible operator. Then an operator $T\in\mathcal{B(H)}$ is said to be of $\mathcal{C}_{A}$-class if the operator valued sequence $\{A_{n}\}_{n\geq 1}$, where

[TABLE]

admits unitary dilation. The only reference we could find regarding this notion is [25]. In particular we couldn’t find part (iii) of the following Theorem in the literature.

Theorem 4.1**.**

Let $A\in\mathcal{B(H)}$ be a positive invertible operator and the operator $T\in\mathcal{B(H)}.$ Then the following are equivalent:

• (i)

$T\in\mathcal{C}_{A}$-class.

• (ii)

For every $N\in\mathbb{N}$ and $c_{1},\cdots,c_{N}\in\mathbb{C}$, we have

[TABLE]

where $\zeta_{A}(n)$ is defined as follows:

[TABLE]

• (iii)

$T$* satisfies the following:*

[TABLE]

Proof.

In view of Theorem 3.1, it is immediate to see (i) and (ii) are equivalent. Now, we claim that (i) and (iii) are equivalent.

(i)$\implies$ (iii): Let ${\bf A}=\{A_{n}\}_{n\geq 0}$, where $A_{n}=A^{-\frac{1}{2}}T^{n}A^{-\frac{1}{2}}$ for $n\geq 1$ and $A_{0}=I.$ Assume that $T\in\mathcal{C}_{A}.$ Then by Theorem 3.1, $P_{\bf A}(\overline{z})\geq 0$ for all $z\in\mathbb{D}.$ In other words,

[TABLE]

It follow that

[TABLE]

Let $h\in\mathcal{H},$ then for all $z\in\mathbb{D},$ we have

[TABLE]

Since \left\langle(I-zT)h,\big{(}(I-zT)^{-1}+(I-\overline{z}T^{*})^{-1}\big{)}(I-zT)h\right\rangle=2Re\langle(I-zT)h,h\rangle, therefore

[TABLE]

(iii)$\implies$ (i): Assume that $\left\langle(I-zT)h,(A-2I)(I-zT)h\right\rangle+2{\rm Re}\langle(I-zT)h,h\rangle\geq 0\text{~{}~{}~{}for all~{}}|z|<1,h\in\mathcal{H}.$ Then by reverse computation, we can show that $P_{\bf A}(\overline{z})\geq 0$ for all $z\in\mathbb{D}.$ Hence by Theorem 3.1, $T\in\mathcal{C}_{A}.$ ∎

In the next result, employing standard techniques we provide another necessary and sufficient criterion in terms of positive maps. As an immediate application of Theorem 4.2, we recover a result of V. Istrăţescu [16] that if $0\leq B\leq A,$ and $A,B$ are invertible, then $\mathcal{C}_{B}\subseteq\mathcal{C}_{A}.$

Theorem 4.2**.**

Let $A\in\mathcal{B(H)}$ be a positive invertible operator. Then an operator $T\in\mathcal{B(H)}$ is in $\mathcal{C}_{A}$-class if and only if the map $\varphi:\mathcal{S}\subseteq C(\mathbb{T})\to\mathcal{B(H)}$ defined by

[TABLE]

is a positive map.

Proof.

Let $T\in{\mathcal{C}}_{A}.$ Now, let $f(e^{i\theta})=\sum\limits_{n=-N}^{N}a_{n}e^{in\theta}\in\mathcal{S}$ be strictly positive. Then applying Riesz-Fejer theorem, we see that $f(e^{i\theta})=\sum\limits_{0\leq\ell,k\leq N}\overline{c_{\ell}}c_{k}e^{i(k-\ell)\theta}$ for some $c_{0},\ldots,c_{N}\in\mathbb{C}.$ This implies that

[TABLE]

As $T\in\mathcal{C}_{A},$ by Theorem 4.1, we have $\sum\limits_{\ell,k=0}^{N}\overline{c_{\ell}}c_{k}\zeta_{A}(k-\ell)\geq 0.$ Therefore $\varphi_{A}(f(z))\geq 0$ whenever $f(z)$ is strictly positively element in $\mathcal{S}.$ Now, let $g\in\mathcal{S}$ be positive. Then $g+\varepsilon\cdot 1$ is strictly positive for any $\epsilon>0.$ Then $\varphi_{A}(g)+\varepsilon A=\varphi_{A}(g+\varepsilon\cdot 1)\geq 0$ for all $\varepsilon>0.$ Taking limit $\epsilon\to 0,$ we obtain $\varphi_{A}(g)\geq 0.$ Hence the map $\varphi_{A}$ is positive.

Conversely, let $\varphi_{A}$ be a positive map. In view of Theorem 4.1, it is enough to prove that $\sum\limits_{\ell,k=0}^{N}\overline{c_{\ell}}c_{k}\zeta_{A}(k-\ell)\geq 0.$ Notice that $\sum\limits_{\ell,k=0}^{N}\overline{c_{\ell}}c_{k}\zeta_{A}(k-l)=A^{-\frac{1}{2}}\varphi_{A}(\sum\limits_{0\leq\ell,k\leq N}\overline{c_{\ell}}c_{k}e^{i(k-\ell)\theta})A^{-\frac{1}{2}}.$ Since $\varphi_{A}$ is positive and $\sum\limits_{0\leq\ell,k\leq N}\overline{c_{\ell}}c_{k}e^{i(k-\ell)\theta}\geq 0,$ therefore $\sum\limits_{\ell,k=0}^{N}\overline{c_{\ell}}c_{k}\zeta_{A}(k-l)\geq 0.$ This completes the proof. ∎

5. Concrete isometric and unitary dilations for a subclass of $\mathcal{C}_{A}$-class operators

Let $A\in\mathcal{B(H)}$ be a positive invertible operator. Recall that an operator $T$ is in $\mathcal{C}_{A}$-class means that the sequence $\{A_{n}\}_{n\geq 1}$ given by

[TABLE]

admits unitary dilation. We do not know how to write down the dilation of $\mathcal{C}_{A}$-class operators in general. Here we do it for a subclass.

First we write down an isometric dilation for $\{A^{-\frac{1}{2}}T^{n}A^{\frac{1}{2}}\}_{n\geq 1}$. Let $A,C\in\mathcal{B(H)}$ be commuting pair such that $A\geq 0$ with $\|C\|\leq 1.$ Suppose,

[TABLE]

Here $I+A(A-2I)C^{*}C\geq 0$ as $C$ is a contraction and $I+A(A-2I)C^{*}C=I-C^{*}C+(A-I)^{2}C^{*}C$. We are assuming that it is invertible. In such a case, we will see that $T$ is in $\mathcal{C}_{A}$-class and we can explicitly write down isometric and unitary dilations of $T$.

Consider $A,C$ as above. It is convenient to have some notation. Take $B=(I+A(A-2I)C^{*}C)^{-\frac{1}{2}},D=(I-C^{*}C)^{\frac{1}{2}},D_{*}=(I-CC^{*})^{\frac{1}{2}}.$ Therefore, $T=ABDC.$ Now we take the following sequence $\{T_{n}\}_{n\geq 0}$ of bounded operators defined by

[TABLE]

Our aim is to show that the sequence $\{T_{n}\}_{n\geq 0}$ admits an isometric dilation. In other words, there exists an isometry $V$ on some Hilbert space $\mathcal{K}\supseteq\mathcal{H}$ such that $T_{n}=P_{\mathcal{H}}V^{n}|_{\mathcal{H}}$ for all $n\geq 0.$ In addition, we want to find an $V$ with explicit block structure.

We observe that since $A$ is positive and commutes with $C$, it also commutes with $C^{*}$. It follows that $A$ commutes with also $B,D,D_{*}$. Consequently we get $BD=DB$. On the other hand, we see that

[TABLE]

As both $D$ and $D_{\ast}$ are positive operators, $D_{\ast}C=CD.$ Furthermore,

[TABLE]

Therefore,

[TABLE]

We first obtain a partial isometric dilation on $\mathcal{H}\oplus\mathcal{H}.$

Lemma 5.1**.**

Define $R$ on $\mathcal{H}\oplus\mathcal{H}$ by:

[TABLE]

Then $R$ is a partial isometry and

[TABLE]

Proof.

Making use of the commutation relations observed above, by direct computation we see that

[TABLE]

Clearly, $R^{*}R$ is a self-adjoint operator and

[TABLE]

Therefore $R^{*}R$ is a projection and $R$ is a partial isometry. Next, we compute $R^{2}$ as follows:

[TABLE]

Therefore, $P_{\mathcal{H}}R^{2}|P_{\mathcal{H}}=A(BDC)^{2}=A^{-\frac{1}{2}}T^{2}A^{-\frac{1}{2}}.$ Hence by induction, we conclude that $P_{\mathcal{H}}R^{n}|_{\mathcal{H}}=A^{-\frac{1}{2}}T^{n}A^{-\frac{1}{2}}.$ ∎

Let $D_{R}=(1-R^{*}R)^{\frac{1}{2}}$ be the defect operator associate the partial isometry. Since $(1-R^{*}R)$ is a projection, $(1-R^{*}R)^{\frac{1}{2}}=(1-R^{*}R).$ Therefore, $D_{R}=\begin{bmatrix}D^{2}&-C^{*}D_{*}\\ -D_{*}C&CC^{*}\end{bmatrix}.$

Remark 5.2**.**

Using the Schäffer construction of $R,$ and Proposition 5.1, we can provide directly explicit form of isometric dilation of the moment sequence $\{A^{-\frac{1}{2}}T^{n}A^{-\frac{1}{2}}\}$ associated to the $\mathcal{C}_{A}$-class. Let $\tilde{V}:\ell^{2}(\mathcal{H}\oplus\mathcal{H})\to\ell^{2}(\mathcal{H}\oplus\mathcal{H})$ be the isometry given by

[TABLE]

Recalling Schäffer construction of $R$, it is clear that $P_{\mathcal{H}}\tilde{V}^{n}|_{\mathcal{H}}=P_{\mathcal{H}}R^{n}|_{\mathcal{H}}=A^{-\frac{1}{2}}T^{n}A^{-\frac{1}{2}}.$

Here is an alternative construction of an isometric dilation which is simpler looking.

Theorem 5.3**.**

Take $\mathcal{K}=\mathcal{H}^{\oplus 3}\oplus\ell^{2}(\mathcal{H}).$ Consider $V$ on $\mathcal{K}$ defined by

[TABLE]

Then $V$ is an isometric dilation of $\{A_{n}\}_{n\geq 0}.$

Proof.

The dilation property follows from the Lemma 5.1. The isometric property of $V$ is clear once we observe that

[TABLE]

∎

In the last theorem there is no claim of minimality. To get the minimal isometric dilation we need to identify the minimal dilation space $\bigvee\limits_{n=0}^{\infty}V^{n}\mathcal{H}.$ This we do here as a proposition.

Proposition 5.4**.**

Let $V$ be the minimal isometric dilation defined in Theorem 5.3. Then the minimal dilation space $\bigvee\limits_{n=0}^{\infty}V^{n}\mathcal{H}$ is of the form

[TABLE]

where $\mathcal{H}_{0}:=\mathcal{H},\mathcal{H}_{1}:=V\mathcal{H}\ominus\mathcal{H}.$ Moreover,

[TABLE]

and

[TABLE]

for $n\geq 2,$ where $P$ is the projection onto $\ker(I-A)BDC.$

Proof.

First note that $\mathcal{H}\bigvee V\mathcal{H}=\mathcal{H}\oplus(V\mathcal{H}\ominus\mathcal{H}).$ Let $h\in\mathcal{H},$ then notice that

[TABLE]

Then it is immediate to see that $V\mathcal{H}\ominus\mathcal{H}=\{(0\oplus(A-I)CBCh\oplus Dh\oplus 0\oplus 0\oplus\cdots):h\in\mathcal{H}\}.$ Next, we shall find $V^{2}\mathcal{H}\ominus(\mathcal{H}\bigvee V\mathcal{H}).$ First, we write $V^{2}h$ as

[TABLE]

Obviously, the first and second terms belong to $\mathcal{H}$ and $V\mathcal{H}\ominus\mathcal{H}$ respectively. We want to understand the last term of this equation. We claim that

[TABLE]

Notice that \big{\langle}\big{(}0\oplus(A-I)CBCg\oplus Dg\oplus 0\oplus 0\oplus\cdots\big{)},\big{(}0\oplus 0\oplus(I-A)BCPh\oplus Dh\oplus 0\oplus\cdots\big{)}\big{\rangle}=0 as $(I-A)BDCPh=0.$ Therefore, we have

[TABLE]

Now, it is sufficient to prove that \big{(}0\oplus 0\oplus(I-A)BC(I-P)h\oplus 0\oplus\cdots\big{)}\in V\mathcal{H}\ominus\mathcal{H} for all $h\in\mathcal{H}.$ To see this, suppose

[TABLE]

for all $g\in\mathcal{H}.$ This implies that for all $g\in\mathcal{H},$

[TABLE]

as $(I-A)BDC(I-P)h=0.$ It follows that $(I-A)BDCh=0$ and $h\in{\rm range}{P}.$ Therefore, we have \big{(}0\oplus 0\oplus(I-A)BC(I-P)h\oplus 0\oplus\cdots\big{)}=0 under the condition defined Equation (36). Consequently, we get \big{(}0\oplus 0\oplus(I-A)BC(I-P)h\oplus 0\oplus\cdots\big{)}\in V\mathcal{H}\ominus\mathcal{H} for all $h\in\mathcal{H}.$ Hence, we have \big{\{}\big{(}0\oplus 0\oplus(I-A)BCPh\oplus Dh\oplus 0\oplus\cdots\big{)}:h\in\mathcal{H}\big{\}}=V^{2}\mathcal{H}\ominus(\mathcal{H}\vee V\mathcal{H}). Now it is easy to see that

[TABLE]

By induction, we can prove that

[TABLE]

∎

Now we explicitly write down a unitary dilation for the sequence $\{T_{n}\}_{n\geq 0}$ as a

Theorem 5.5**.**

Let $\mathcal{K}=\ell^{2}(\mathcal{H})\oplus\mathcal{H}^{\oplus 3}\oplus\ell^{2}(\mathcal{H})$ and $U$ acting on $\mathcal{K}$ is defined by

[TABLE]

where $BDC$ that appear in the bold font is the $(00)$ entry of $U,$ and the $B_{*}$ is given by

[TABLE]

Then $U$ is a unitary dilation of $\{T_{n}\}_{n\geq 0}.$

Proof.

Consider the block operator matrix $M$ defined by:

[TABLE]

It is straight forward to verify that

[TABLE]

This clearly implies that $U^{*}U=UU^{*}=I.$ Finally, we obtain the desired claim

[TABLE]

These completes the proof. ∎

As we discussed earlier, minimal dilation space of the unitary dilation is of the form $\mathcal{K}=\bigvee^{\infty}\limits_{n=-\infty}U^{n}\mathcal{H}.$ The minimal dilation space can be written as $\mathcal{K}_{+}\bigvee\mathcal{K}^{-}=\mathcal{K},$ where $\mathcal{K}_{+}=\bigvee^{\infty}\limits_{n=0}U^{n}\mathcal{H},$ and $\mathcal{K}^{-}=\bigvee^{-\infty}\limits_{n=0}U^{n}\mathcal{H}.$ Since $U|_{\mathcal{K}_{+}}$ is the isometry $V$ constructed before, the decomposition of $\mathcal{K}_{+}$ is clear.

Remark 5.6**.**

$\mathcal{K}_{-}$* can be decomposed as*

[TABLE]

$\text{ where }\mathcal{H}_{-1}:=U^{*}\mathcal{H}\ominus\mathcal{H},\mathcal{H}_{-n}:=U^{*n}\mathcal{H}\ominus\bigvee\limits_{m=0}^{n-1}U^{*m}\mathcal{H}\text{ for }n\geq 2.$* Moreover,*

[TABLE]

[TABLE]

and $Q$ is the projection onto $\ker(A-1)B^{\frac{1}{2}}C^{*}DB.$

5.1. Special case ($\mathcal{C}_{\rho}$-class)

The case when $A$ is a positive scalar $\rho,$ corresponds to the $\mathcal{C}_{\rho}$- class of operators. This class is very rich and widely studied. However, as far as we know explicit block operator description of isometric and unitary dilations of operators this class is not found in the literature except for $\rho=2$. An operator $T$ is in $\mathcal{C}_{2}$ class if and only if $w(T)\leq 1,$ where $w(T)$ denote the numerical range of $T$. Unitary dilation of this class was exhibited by T. Ando [6] and coincides with the following construction (with $\rho=2$). We recall from Durzst [14] that $T\in\mathcal{C}_{\rho}(\rho>0)$ if and only if $T$ is of the form:

[TABLE]

where $\rho>0,C$ is a contraction and $D=(I-C^{*}C)^{\frac{1}{2}}.$ In this case we may write down isometric and unitary dilations as above. We observe that $B=(I+\rho(\rho-2)C^{*}C)^{-\frac{1}{2}},B_{*}=(I+\rho(\rho-2)CC^{*})^{-\frac{1}{2}}$ and $T_{n}=\rho^{n-1}(BDC)^{n}=\frac{(\rho BDC)^{n}}{\rho}=\frac{T^{n}}{\rho}.$ Moreover, for the case $\rho=2,$ being $B=B^{*}=I,$ the operators $B,B^{*}$ will not play any role in unitary and isometric dilation of $\mathcal{C}_{2}$-class.

Question 5.7**.**

How to write block decompositions for isometric and unitary dilations of general $\mathcal{C}_{A}$-class operators?

Acknowledgements: The first author is funded by the J C Bose Fellowship JBR/2021/000024 of SERB(India). The second author is supported by the NBHM postdoctoral fellowship, Department of Atomic Energy (DAE), Government of India (File No. 0204/1(4)/2022/ R&D-II/1198). The third author is funded by the Startup Research Grant (File No. SRG/2022/001795) of SERB, India. We also acknowledge Stat-Math Unit of the Indian Statistical Institute Bangalore Centre for providing excellent research environment.