Structure theorems for operators associated with two domains related to $\mu$-synthesis

A commuting tuple of $n$ operators $(S_1,\dots ,S_{n-1},P)$ defined on a Hilbert space $\mathcal{H}$, for which the closed symmetrized polydisc $${\mathrm{\Gamma }}_n=\{(\sum _{i=1}^n{z}_i,\sum _{1\le i<j\le n}{z}_i{z}_j,\dots ,\prod _{i=1}^n{z}_i):\mathrm{\mid }{z}_i\mathrm{\mid }\le 1,i=1,\dots ,n\}$$ is a spectral set is called a ${\mathrm{\Gamma }}_n$-contraction. Also a triple of commuting operators $(A,B,P)$ for which the closed tetrablock $\bar{\mathbb{E}}$ is a spectral set is called an $\mathbb{E}$-contraction, where $$\mathbb{E}=\{(x_1,x_2,x_3)\in {\mathbb{C}}^3:1-z{x}_1-w{x}_2+zw{x}_3\mathrm{\ne }0\mathrm{\forall }z,w\in \bar{\mathbb{D}}\}\mathrm{.}$$ 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 ${\mathrm{\Gamma }}_n$-contractions and $\mathbb{E}$-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 ${\mathrm{\Gamma }}_n$-contraction or an $\mathbb{E}$-contraction is played by positivity of some certain operator pencils and the existence of a unique operator tuple associated with a ${\mathrm{\Gamma }}_n$-contraction or an $\mathbb E$-contraction.