Operators on Hilbert Space having ${\Gamma }_{E(3;3;1,1,1)}$ and ${\Gamma }_{E(3;2;1,2)}$ as Spectral Sets

Abstract

A $7$-tuple of commuting bounded operators $\text{T}=(T_1,\dots ,T_7)$ on a Hilbert space $\mathcal{H}$ is called a \textit{${\Gamma }_{E(3;3;1,1,1)}$-contraction} if ${\Gamma }_{E(3;3;1,1,1)}$ is a spectral set for $\text{T}.$ 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 $S_i{\tilde{S}}_j={\tilde{S}}_j{S}_i$ for $1⩽i⩽3$ and $1⩽j⩽2$. We say that $\text{S}=(S_1,S_2,S_3,{\tilde{S}}_1,{\tilde{S}}_2)$ is a ${\Gamma }_{E(3;2;1,2)}$-contraction if ${\Gamma }_{E(3;2;1,2)}$ is a spectral set for $\text{S}$. We derive various properties of ${\Gamma }_{E(3;3;1,1,1)}$-contractions and ${\Gamma }_{E(3;2;1,2)}$-contractions and establish a relationship between them. We discuss the fundamental equations for ${\Gamma }_{E(3;3;1,1,1)}$-contractions and ${\Gamma }_{E(3;2;1,2)}$-contractions. We explore the structure of ${\Gamma }_{E(3;3;1,1,1)}$-unitaries and ${\Gamma }_{E(3;2;1,2)}$-unitaries and elaborate on the relationship between them. We also study various properties of ${\Gamma }_{E(3;3;1,1,1)}$-isometries and ${\Gamma }_{E(3;2;1,2)}$-isometries. We discuss the Wold Decomposition for a ${\Gamma }_{E(3;3;1,1,1)}$-isometry and a ${\Gamma }_{E(3;2;1,2)}$-isometry. We further outline the structure theorem for a pure ${\Gamma }_{E(3;3;1,1,1)}$-isometry and a pure ${\Gamma }_{E(3;2;1,2)}$-isometry.