Generalized Foguel-Hankel Operators

TL;DR

This paper introduces a broader class of Foguel-Hankel operators using multiplication operators on Hardy space, analyzing their power boundedness and similarity to contractions, with new insights into Peller's and Kreiss conditions.

Contribution

It generalizes Foguel-Hankel operators by replacing the shift with multiplication operators and explores the conditions for power boundedness and similarity to contractions.

Findings

Peller's condition is sufficient for power boundedness.

Peller's condition is not necessary for power boundedness in general.

Similarity to a contraction is equivalent to the Kreiss condition for the Hilbert matrix case.

Abstract

In this paper we introduce a more general class of Foguel-Hankel operators, where the unilateral shift on $\ell^2(\mathbb{N}) $ is replaced by a general multiplication operator on the Hardy space $H^2$ . We prove that Peller's condition is sufficient for the operator to be power bounded, but in general it is not necessary. When the Hankel matrix is the Hilbert matrix, we prove that being similar to a contraction is equivalent to the (a priori) weaker Kreiss condition.