Generalized Foguel-Hankel Operators

TL;DR

This work generalizes Foguel-Hankel operators by replacing the shift with a multiplication operator $M_\varphi$ on $H^2$, forming the generalized Foguel-Hankel operator $\Gamma_{f,\varphi}$. It shows that if $f'$ is a Bloch function and $\varphi$ is a holomorphic self-map of the disk, then $\Gamma_{f,\varphi}$ is power bounded, while in general the Kreiss condition is not equivalent to power boundedness. In the Hilbert-matrix subcase, similarity to a contraction, the Kreiss condition, and the boundary-modulus condition $\limsup_{r\to1^-}|\varphi(r)|<1$ become equivalent, with the outcome determined by $|\varphi|$ on $(0,1)$. The paper also develops a tractable polynomial calculus for $\Gamma_{f,\varphi}$, leverages Carleson/Luecking embedding techniques, and lays groundwork for further characterizations of boundedness and similarity phenomena in this generalized setting.

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.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Theorem 4
• Lemma 2
• proof
• Theorem 5
• proof : Proof of Theorem \ref{['thm:FH_Kreiss']}