Operators associated with the pentablock and their relations with biball and symmetrized bidisc

TL;DR

This paper characterizes operators related to the pentablock, establishing their decompositions, dilations, and relations with biball and symmetrized bidisc domains in operator theory.

Contribution

It introduces new characterizations and decompositions for $bP$-contractions, $bP$-isometries, and $bP$-unitaries, and constructs explicit dilations connecting these operators with geometric domains.

Findings

Every $bP$-isometry admits a Wold type decomposition.

Every $bP$-contraction can be orthogonally decomposed into a $bP$-unitary and a non-unitary part.

Explicit construction of a dilation from $bP$-contraction to $bP$-isometry.

Abstract

A commuting triple of Hilbert space operators $(A,S,P)$ is said to be a \textit{$\mathbb{P}$-contraction} if the closed pentablock $\overline{\mathbb P}$ is a spectral set for $(A,S,P)$, where \[ \mathbb{P}:=\left\{(a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0))\ : \ A_0=[a_{ij}]_{2 \times 2} \; \; \& \;\; \|A_0\| <1 \right\} \subseteq \mathbb{C}^3. \] A commuting triple of normal operators $(A, S, P)$ acting on a Hilbert space is said to be a \textit{$\mathbb P$-unitary} if the Taylor-joint spectrum $\sigma_T(A, S, P)$ of $(A, S, P)$ is contained in the distinguished boundary $b\mathbb{P}$ of $\PC$. Also, $(A, S , P)$ is called a \textit{$\mathbb P$-isometry} if it is the restriction of a $\mathbb P$-unitary $(\hat A, \hat S, \hat P)$ to a joint invariant subspace of $\hat A, \hat S, \hat P$. We find several characterizations for the $\mathbb P$-unitaries and $\mathbb P$-isometries. We show that every $\mathbb P$-isometry admits a Wold type decomposition that splits it into a direct sum of a $\mathbb P$-unitary and a pure $\mathbb P$-isometry. Moving one step ahead we show that every $\mathbb P$-contraction $(A,S,P)$ possesses a canonical decomposition that orthogonally decomposes $(A,S,P)$ into a $\mathbb P$-unitary and a completely non-unitary $\mathbb P$-contraction. We find a necessary and sufficient condition such that a $\mathbb P$-contraction $(A, S, P)$ dilates to a $\mathbb P$-isometry $(X, T, V)$ with $V$ being the minimal isometric dilation of $P$. Then we show an explicit construction of such a conditional dilation. We show interplay between operator theory on the following three domains: the pentablock, the biball and the symmetrized bidisc.