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

A commuting tuple of n operators (S 1 , ... , S n − 1 , P) defined on a Hilbert space H , for which the closed symmetrized polydisc Γ n = { (∑ i = 1 n z i , ∑ 1 ≤ i < j ≤ n z i z j , ... , ∏ i = 1 n z 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 = { (x 1 , x 2 , x 3) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 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. [ABSTRACT FROM AUTHOR]