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

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{${\Gamma }_{E(3;3;1,1,1)}$-contraction} if ${\Gamma }_{E(3;3;1,1,1)}$ 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 $1\le i\le 3$ and $1\le j\le 2$. The tuple $\mathbf{S}=(S_1,S_2,S_3,{\tilde{S}}_1,{\tilde{S}}_2)$ is called a \textit{${\Gamma }_{E(3;2;1,2)}$-contraction} if ${\Gamma }_{E(3;2;1,2)}$ is a spectral set for $\mathbf{S}$. In this paper, we establish the existence and uniqueness of the fundamental operators associated with ${\Gamma }_{E(3;3;1,1,1)}$-contractions and ${\Gamma }_{E(3;2;1,2)}$-contractions. Furthermore, we obtain a Beurling-Lax-Halmos type representation for invariant subspaces corresponding to a pure ${\Gamma }_{E(3;3;1,1,1)}$-isometry and a pure ${\Gamma }_{E(3;2;1,2)}$-isometry. We also construct a conditional dilation for a ${\Gamma }_{E(3;3;1,1,1)}$-contraction and a ${\Gamma }_{E(3;2;1,2)}$-contraction and develop an explicit functional model for a certain subclass of these operator tuples. Finally, we demonstrate that every ${\Gamma }_{E(3;3;1,1,1)}$-contraction (respectively, ${\Gamma }_{E(3;2;1,2)}$-contraction) admits a unique decomposition as a direct sum of a ${\Gamma }_{E(3;3;1,1,1)}$-unitary (respectively, ${\Gamma }_{E(3;2;1,2)}$-unitary) and a completely non-unitary ${\Gamma }_{E(3;3;1,1,1)}$-contraction (respectively, ${\Gamma }_{E(3;2;1,2)}$-contraction).