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

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