Common reducing subspaces and decompositions of contractions

Abstract

Abstract A commuting triple of Hilbert space operators ( A , B , P ) {(A,B,P)} , for which the closed tetrablock 𝔼 ¯ {\overline{\mathbb{E}}} is a spectral set, is called a tetrablock-contraction or simply an 𝔼 {\mathbb{E}} -contraction, where 𝔼 = { ( x 1 , x 2 , x 3 ) ∈ ℂ 3 : 1 - x 1 ⁢ z - x 2 ⁢ w + x 3 ⁢ z ⁢ w ≠ 0 ⁢  whenever  ⁢ | z | ≤ 1 ⁢  and  ⁢ | w | ≤ 1 } ⊂ ℂ 3 , \mathbb{E}=\bigl{\{}(x_{1},x_{2},x_{3})\in\mathbb{C}^{3}:1-x_{1}z-x_{2}w+x_{3}% zw\neq 0\text{ whenever }\lvert z\rvert\leq 1\text{ and }\lvert w\rvert\leq 1% \bigr{\}}\subset\mathbb{C}^{3}, is a polynomially convex domain which is naturally associated with the μ-synthesis problem. By applications of the theory of 𝔼 {\mathbb{E}} -contractions, we obtain the following main results in this article: (i) It is well known that any number of doubly commuting isometries admit a Wold-type decomposition. Also, examples from the literature show that such decomposition does not hold in general for commuting isometries. We generalize this result to any number of doubly commuting contractions. Indeed, we present a canonical decomposition to any family of doubly commuting contractions 𝒰 = { T λ : λ ∈ Λ } {\mathcal{U}=\{T_{\lambda}:\lambda\in\Lambda\}} acting on a Hilbert space ℋ {\mathcal{H}} and show that ℋ {\mathcal{H}} continues to decompose orthogonally into joint reducing subspaces of 𝒰 {\mathcal{U}} until all members of 𝒰 {\mathcal{U}} split together into unitaries and completely non-unitary (c.n.u.) contractions. Naturally, as a special case we achieve an analogous Wold decomposition for any number of doubly commuting isometries. (ii) Further, we show that a similar canonical decomposition is possible for any family of doubly commuting c.n.u. contractions, where all members jointly orthogonally decompose into pure isometries (i.e. unilateral shifts) and completely non-isometry (c.n.i.) contractions. (iii) We give an alternative proof to the canonical decomposition of an 𝔼 {\mathbb{E}} -contraction and apply that to establish independently the following result due to Eschmeier: for any finitely many commuting contractions T 1 , … , T n {T_{1},\dots,T_{n}} acting on a Hilbert space ℋ {\mathcal{H}} , the space ℋ {\mathcal{H}} admits an orthogonal decomposition ℋ = ℋ 1 ⊕ ℋ 2 {\mathcal{H}=\mathcal{H}_{1}\oplus\mathcal{H}_{2}} , where ℋ 1 {\mathcal{H}_{1}} is the maximal common reducing subspace for T 1 , … , T n {T_{1},\dots,T_{n}} such that T 1 | ℋ 1 , … , T n | ℋ 1 {T_{1}\lvert_{\mathcal{H}_{1}},\dots,T_{n}\lvert_{\mathcal{H}_{1}}} are unitaries.