Operators near completely polynomially dominated ones and similarity problems

TL;DR

This paper investigates the similarity of Hilbert space operators near polynomially dominated operators, providing new results on their structure, including a refined Banach space version of Rota's theorem and insights into CAR-valued Foguel-Hankel operators.

Contribution

It introduces conditions under which near polynomially dominated operators are similar to operators dominated by a direct sum, advancing understanding of operator similarity and polynomial domination.

Findings

Operators near polynomially dominated are similar to those dominated by a direct sum.

Refined Banach space version of Rota's similarity theorem established.

Partial solutions to problems on CAR-valued Foguel-Hankel operators provided.

Abstract

Let T and C be two Hilbert space operators. We prove that if T is near, in a certain sense, to an operator completely polynomially dominated with a finite bound by C, then T is similar to an operator which is completely polynomially dominated by the direct sum of C and a suitable weighted unilateral shift. Among the applications, a refined Banach space version of Rota similarity theorem is given and partial answers to a problem of K. Davidson and V. Paulsen are obtained. The latter problem concerns CAR-valued Foguel-Hankel operators which are generalizations of the operator considered by G. Pisier in his example of a polynomial bounded operator not similar to a contraction.