Weak compactness and essential norms of integration operators

TL;DR

This paper investigates the properties of integration operators on certain function spaces, establishing conditions under which weak compactness and compactness coincide, and providing estimates for essential norms.

Contribution

It proves that on $H^1$ and $BMOA$, weak compactness of $T_g$ is equivalent to compactness, and offers general estimates for essential and weak essential norms.

Findings

Weak compactness equals compactness for $T_g$ on $H^1$ and $BMOA$.

Provides estimates for essential and weak essential norms of $T_g$ on $H^p$ and $BMOA$.

Answers a question of Siskakis and Zhao regarding $BMOA$.

Abstract

Let $g$ be an analytic function on the unit disc and consider the integration operator of the form $T_g f(z) = \int_0^z fg'\,d\zeta$. We show that on the spaces $H^1$ and $BMOA$ the operator $T_g$ is weakly compact if and only if it is compact. In the case of $BMOA$ this answers a question of Siskakis and Zhao. More generally, we estimate the essential and weak essential norms of $T_g$ on $H^p$ and $BMOA$.