On similarity to contractions of class $C_{\cdot 0}$ with finite defects

TL;DR

This paper provides criteria for when a bounded linear operator on a Hilbert space is similar to a contraction of class C_{0} with finite defects, based on invariant subspaces and spectral properties.

Contribution

It establishes a new criterion linking the similarity to C_{0} contractions with finite defects to the structure of invariant subspaces and spectral conditions of the operator.

Findings

Operator T is similar to a C_{0} contraction if minimal invariant subspaces span the space.

A sufficient condition involves spectral multiplicity and Blaschke product conditions.

The paper characterizes similarity via invariant subspace structure and spectral properties.

Abstract

A criterion on the similarity of a (bounded, linear) operator $T$ on a (complex, separable) Hilbert space $\mathcal H$ in terms of shift-type invariant subspaces of $T$ to a contraction of class $C_{\cdot 0}$ with finite unequal defects is given. Namely, $T$ is similar to such a contraction if and only if the minimal quantity of (closed) invariant subspaces $\mathcal M$ of $T$ such that the restriction $T|_{\mathcal M}$ of $T$ on $\mathcal M$ is similar to the simple unilateral shift, whose linear span is $\mathcal H$, is finite. A sufficient condition for the similarity of an absolutely continuous polynomially bounded operator $T$ to a contraction of class $C_{\cdot 0}$ with finite equal defects is given. Namely, $T$ is similar to such a contraction if the (spectral) multiplicity of $T$ is finite and $B(T)=\mathbb O$, where $B$ is a finite product of Blaschke products with simple zeros satisfying the Carleson interpolating condition (a Carleson--Newman product).