Common reducing subspaces and decompositions of contractions

TL;DR

This paper investigates the structure of contractions in Hilbert spaces through the lens of $ extit{E}$-contractions, revealing new decomposition results by leveraging the spectral set properties of the tetrablock domain.

Contribution

It introduces new decomposition results for contractions based on the theory of $ extit{E}$-contractions and their spectral set properties related to the tetrablock domain.

Findings

Derived new decomposition theorems for contractions

Linked spectral set properties to operator decompositions

Applied $ extit{E}$-contraction theory to solve decomposition problems

Abstract

A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set, is called a \textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction}, where \[ \mathbb E=\{(x_1,x_2,x_3)\in \mathbb C^3:\, 1-x_1z-x_2w+x_3zw \neq 0 \quad \text{ whenever } \; |z|\leq 1, \; \; |w|\leq 1 \} \subset \mathbb C^3, \] is a polynomially convex domain which is naturally associated with the $\mu$-synthesis problem. By applications of the theory of $\mathbb E$-contractions, we obtain several results on decompositions of contractions.