Operator moment dilations as block operators

TL;DR

This paper investigates the operator moment dilation problem, seeking dilations $B$ that realize a given operator sequence $\{A_n\}$ via $A_n = P_{\mathcal{H}} B^n|_{\mathcal{H}}$. It develops both abstract and constructive frameworks: self-adjoint dilations are characterized by Hankel-positivity conditions on $H_n$ (and variants), while unitary/isometric dilations are captured through Poisson-kernel criteria and CP-map techniques. It then introduces the $\mathcal{C}_{A}$-class to study operator sequences via $A$-weighted dilations and provides equivalent positivity/CP-characterizations, including spectral constraints. Finally, it presents explicit, block-operator dilations for a subclass of $\mathcal{C}_{A}$-class operators, including Schäffer-type representations and concrete templates for $T$ that yield isometric and unitary dilations, as well as a special scalar case $A=\rho$ (with $\rho=2$ recovering Ando’s dilation). These results deliver transparent, tractable block representations that crystallize the operator moment problem and connect it with classical dilation theory and Jacobi-type structures.

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 $ρ$-dilations for positive scalars. Here the special cases $ρ=1$ and $ρ=2$ correspond to Schäffer representation for contractions and Ando representation for operators with numerical radius not more than one respectively.

Theorems & Definitions (43)

• Definition 1.1: Dilation of operator sequences
• Theorem 1.3
• proof
• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Remark 2.4
• Lemma 2.5