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

A $7$-tuple of commuting bounded operators $\textbf{T} = (T_1, \dots, T_7)$ 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 $\textbf{T}.$ 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 $\Gamma_{E(3; 2; 1, 2)}$-contraction if $\Gamma_{E(3; 2; 1, 2)}$ is a spectral set for $\textbf{S}$. 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.