Contractions with Polynomial Characteristic Functions II. Analytic Approach

2016/04/19 by Foias, Ciprian, Pearcy, Carl, Sarkar, Jaydeb
#47A20 #47A45 #47A48 #47A56 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1604.05485

Abstract

The simplest and most natural examples of completely nonunitary contractions on separable complex Hilbert spaces which have polynomial characteristic functions are the nilpotent operators. The main purpose of this paper is to prove the following theorem: Let T be a completely nonunitary contraction on a Hilbert space H. If the characteristic function Θ_T of T is a polynomial of degree m, then there exist a Hilbert space M, a nilpotent operator N of order m, a coisometry V₁ ∈ L(ran (I - N N^*) ⊕ M, ran (I - T T^*)), and an isometry V₂ ∈ L(ran (I - T^* T), ran (I - N^* N) ⊕ M), such that Θ_T = V₁ \beginbmatrix Θ_N amp; 0 0 amp; I_M \endbmatrix V₂.

Related

• Contractions with Polynomial characteristic functions I. Geometric approach
• Characteristic functions and joint invariant subspaces
• Characteristic function of a power partial isometry
• Constrained dilation and Γ-contractions
• A new functional model for contractions