Essential norm and integration of a family of weighted composition operators

TL;DR

This paper investigates when the essential norm of an integral of weighted composition operators equals the integral of their essential norms on weighted Bergman spaces, providing conditions and calculations for specific cases.

Contribution

It establishes sufficient and necessary conditions for the interchange of essential norm and integration for families of weighted composition operators on Bergman spaces, including explicit calculations for certain operators.

Findings

Derived a sufficient condition for essential norm interchange

Provided necessary conditions for the equality to hold

Calculated essential norms for specific integral operators like Volterra operators

Abstract

We study the interchange of essential norm and integration of certain families of weighted composition operators acting on the standard weighted Bergman spaces $A^p_\alpha$, where $p>1$ and $\alpha\geq 0$. To be more precise, we give a sufficient condition for $ \|\int u_tC_{\phi_t}\, dt\|_e = \int \| u_tC_{\phi_t}\|_e \, dt $ to hold in terms of geometric properties of $u_t$ and $\phi_t$. We also provide some necessary conditions for the equality to hold and calculate the essential norm of some integral operators such as some Volterra operators.