Structure theorems for operators associated with two domains related to μ-synthesis

A commuting tuple of n operators (S₁,…,S_n−1,P) defined on a Hilbert space H, for which the closed symmetrized polydisc Γ_n={(∑i=1nz_i,∑1≤i<j≤nz_iz_j,…,∏i=1nz_i):|z_i|≤1,i=1,…,n} is a spectral set is called a Γ_n-contraction. Also a triple of commuting operators (A,B,P) for which the closed tetrablock E‾ is a spectral set is called an E-contraction, where E={(x₁,x₂,x₃)∈C³:1−zx₁−wx₂+zwx₃≠0∀z,w∈D‾}. There are several decomposition theorems for contraction operators in the literature due to Sz. Nagy, Foias, Levan, Kubrusly, Foguel and few others which reveal structural information of a contraction. In this article, we obtain analogues of six such major theorems for both Γ_n-contractions and E-contractions. In each of these decomposition theorems, the underlying Hilbert space admits a unique orthogonal decomposition which is provided by the last component P. The central role in determining the structure of a Γ_n-contraction or an E-contraction is played by positivity of some certain operator pencils and the existence of a unique operator tuple associated with a Γ_n-contraction or an E-contraction.