Essential norm and integration of a family of weighted composition operators
David Norrbo
TL;DR
The paper studies when the essential norm of the integral mean of weighted composition operators on weighted Bergman spaces $A^p_α$ commutes with integration, i.e., when $||∫_0^1 S_t dt||_e = ∫_0^1 ||S_t||_e dt$. It introduces admissible families with a common boundary direction and uses dominating approximate evaluation maps to derive exact essential-norm formulas, proving a main result under the condition that $\arg u_t(ξ)$ is constant in $t$ and leveraging the $M_p$ property of $A^p_α$. It provides explicit calculations for Volterra-type operators and generalized Hilbert matrix operators, yielding concrete expressions for their essential norms, and includes a nonunivalent example to illustrate the framework. The results supply geometric and operator-theoretic tools for exact essential-norm computations of integral representations on Bergman spaces, with implications for spectral analysis and $M$-ideals in operator algebras.
Abstract
We study the interchange of essential norm and integration of certain families of weighted composition operators acting on the standard weighted Bergman spaces $A^p_α$, where $p>1$ and $α\geq 0$. To be more precise, we give a sufficient condition for $ \|\int u_tC_{φ_t}\, dt\|_e = \int \| u_tC_{φ_t}\|_e \, dt $ to hold in terms of geometric properties of $u_t$ and $φ_t$. 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.