Volterra-type inner derivations on Hardy spaces

TL;DR

This work characterizes when Volterra-type inner derivations act compactly on operator algebras of Hardy spaces, extending Calkin's classical result to Banach-space settings. The authors reduce the problem to compact intertwining relations for Volterra and composition operators and employ generalized area operators and Carleson-measure criteria to obtain sharp necessary and sufficient conditions. They prove that the $J_g$-inner derivation maps bounded operators into the compact ideal on $\mathcal{H}^p$ iff $g \in \mathrm{VMOA}$, while the $I_g$-inner derivation does so iff $g$ is a scalar, and they derive corresponding compact intertwining criteria for multiplication and composition operators between $\mathcal H^p$ and $\mathcal A_\alpha^q$. The results deepen the understanding of the operator-theoretic structure of Hardy and Bergman spaces and provide precise thresholds (VMOA, Carleson measures, and area operators) governing compactness in these settings.

Abstract

A classical result of Calkin [Ann. of Math. (2) 42 (1941), pp. 839-873] says that an inner derivation $S\mapsto [T,S] = TS-ST$ maps the algebra of bounded operators on a Hilbert space into the ideal of compact operators if and only if $T$ is a compact perturbation of the multiplication by a scalar. In general, an analogous statement fails for operators on Banach spaces. To complement Calkin's result, we characterize Volterra-type inner derivations on Hardy spaces using generalized area operators and compact intertwining relations for Volterra and composition operators. Further, we characterize the compact intertwining relations for multiplication and composition operators between Hardy and Bergman spaces.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Remark
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7