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

Abstract

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 \[ \Gamma_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n}z_iz_j, \dots, \prod_{i=1}^{n}z_i \right) : |z_i|\leq 1, i=1, \dots, n \right\} \] is a spectral set is called a $\Gamma_n$-contraction. Also a triple of commuting operators $(A,B,P)$ for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set is called an $\mathbb E$-contraction, where \[ \mathbb E = \{ (x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3 \neq 0 \quad \forall z, w \in \overline{\mathbb D} \}. \] 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 $\Gamma_n$-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 $\Gamma_n$-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 $\Gamma_n$-contraction or an $\mathbb E$-contraction.