Canonical Decompositions and Conditional Dilations of $Γ_{E(3; 3; 1, 1, 1)}$-Contraction and $Γ_{E(3; 2; 1, 2)}$-Contraction

TL;DR

The paper advances multivariable dilation theory for two families of operator tuples, the $Γ_{E(3; 3; 1, 1, 1)}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions. It establishes the existence and uniqueness of fundamental operators, develops Beurling-Lax-Halmos type representations for invariant subspaces of pure isometries, constructs conditional Schäffer-type dilations, and provides explicit functional models and a canonical decomposition into unitary and completely non-unitary parts. These results yield a coherent framework for understanding spectral-set behavior, dilation, and invariant-subspace structure in this targeted multivariable setting. The work deepens the connection between operator model theory and spectral-set geometry, with potential impact on multivariable dilation questions and related domains such as tetrablock and symmetrized-domain theories.

Abstract

A $7$-tuple of commuting bounded operators $\mathbf{T} = (T_1, \dots, T_7)$ defined on a Hilbert space $\mathcal{H}$ is said to be a \textit{$Γ_{E(3; 3; 1, 1, 1)}$-contraction} if $Γ_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\mathbf{T}$. Let $(S_1, S_2, S_3)$ and $(\tilde{S}_1, \tilde{S}_2)$ be tuples of commuting bounded operators on $\mathcal{H}$ satisfying $S_i \tilde{S}_j = \tilde{S}_j S_i$ for $1 \leq i \leq 3$ and $1 \leq j \leq 2$. The tuple $\mathbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ is called a \textit{$Γ_{E(3; 2; 1, 2)}$-contraction} if $Γ_{E(3; 2; 1, 2)}$ is a spectral set for $\mathbf{S}$. In this paper, we establish the existence and uniqueness of the fundamental operators associated with $Γ_{E(3; 3; 1, 1, 1)}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions. Furthermore, we obtain a Beurling-Lax-Halmos type representation for invariant subspaces corresponding to a pure $Γ_{E(3; 3; 1, 1, 1)}$-isometry and a pure $Γ_{E(3; 2; 1, 2)}$-isometry. We also construct a conditional dilation for a $Γ_{E(3; 3; 1, 1, 1)}$-contraction and a $Γ_{E(3; 2; 1, 2)}$-contraction and develop an explicit functional model for a certain subclass of these operator tuples. Finally, we demonstrate that every $Γ_{E(3; 3; 1, 1, 1)}$-contraction (respectively, $Γ_{E(3; 2; 1, 2)}$-contraction) admits a unique decomposition as a direct sum of a $Γ_{E(3; 3; 1, 1, 1)}$-unitary (respectively, $Γ_{E(3; 2; 1, 2)}$-unitary) and a completely non-unitary $Γ_{E(3; 3; 1, 1, 1)}$-contraction (respectively, $Γ_{E(3; 2; 1, 2)}$-contraction).

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Lemma 2.1: Lemma $2.9,$ Roy
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• proof
• Definition 2.5
• Theorem 2.6