Operators on Hilbert Space having $Γ_{E(3; 3; 1, 1, 1)}$ and $Γ_{E(3; 2; 1, 2)}$ as Spectral Sets
Dinesh Kumar Keshari, Avijit Pal, Bhaskar Paul
TL;DR
This work studies seven- and five-tuples of commuting bounded operators whose joint spectra lie in the generalized tetrablock domains $Γ_{E(3;3;1,1,1)}$ and $Γ_{E(3;2;1,2)}$, establishing a cohesive dilation and model theory for these non-convex spectral sets. It develops a comprehensive framework of contractions, unitaries, and isometries tied to these domains, including fundamental equations $ρ^{(1)}, ρ^{(2)}, ρ^{(3)}$, and operator-function models that connect $Γ_{E(3;3;1,1,1)}$-contractions to $Γ_{E(3;2;1,2)}$-contractions via projections and η-parametrized maps. The paper proves a suite of equivalences for unitary and isometric realizations, provides a detailed Wold decomposition, and gives structure theorems for pure isometries modeled on vector-valued Hardy spaces with operator-valued symbols. These results extend multivariable dilation theory to the generalized tetrablock setting, offering explicit model constructions and bridging the two domains through precise transformation rules and spectral criteria for unitary and isometric cases.
Abstract
A $7$-tuple of commuting bounded operators $\textbf{T} = (T_1, \dots, T_7)$ on a Hilbert space $\mathcal{H}$ is called a \textit{$Γ_{E(3; 3; 1, 1, 1)} $-contraction} if $Γ_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\textbf{T}. $ Let $(S_1, S_2, S_3)$ and $(\tilde{S}_1, \tilde{S}_2)$ be tuples of commuting bounded operators defined on a Hilbert space $\mathcal{H}$ with $S_i\tilde{S}_j = \tilde{S}_jS_i$ for $1 \leqslant i \leqslant 3$ and $1 \leqslant j \leqslant 2$. We say that $\textbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ is a $Γ_{E(3; 2; 1, 2)} $-contraction if $ Γ_{E(3; 2; 1, 2)}$ is a spectral set for $\textbf{S}$. We derive various properties of $Γ_{E(3; 3; 1, 1, 1)}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions and establish a relationship between them. We discuss the fundamental equations for $Γ_{E(3; 3; 1, 1,1 )}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions. We explore the structure of $Γ_{E(3; 3; 1, 1, 1)}$-unitaries and $Γ_{E(3; 2; 1, 2)}$-unitaries and elaborate on the relationship between them. We also study various properties of $Γ_{E(3; 3; 1, 1, 1)}$-isometries and $Γ_{E(3; 2; 1, 2)}$-isometries. We discuss the Wold Decomposition for a $Γ_{E(3; 3; 1, 1, 1)}$-isometry and a $Γ_{E(3; 2; 1, 2)}$-isometry. We further outline the structure theorem for a pure $Γ_{E(3; 3; 1, 1, 1)}$-isometry and a pure $Γ_{E(3; 2; 1, 2)}$-isometry.