Sums of Weighted Differentiation Composition Operators

TL;DR

This paper addresses an interpolation problem in weighted Bergman spaces, providing bounds and characterizations for sums of weighted composition and differentiation operators, including their boundedness and compactness properties.

Contribution

It introduces a new interpolation framework in weighted Bergman spaces and characterizes the boundedness and compactness of related operator sums.

Findings

Bounded norm for interpolation problem independent of points

Characterization of order-boundedness of operator sums

Criteria for compactness of operators into bounded analytic functions

Abstract

We solve an interpolation problem in $A^p_\alpha$ involving specifying a set of (possibly not distinct) $n$ points, where the $k^{\textrm{th}}$ derivative at the $k^{\textrm{th}}$ point is up to a constant as large as possible for functions of unit norm. The solution obtained has norm bounded by a constant independent of the points chosen. As a direct application, we obtain a characterization of the order-boundedness of a sum of products of weighted composition and differentiation operators acting between weighted Bergman spaces. We also characterize the compactness of such operators that map a weighted Bergman space into the space of bounded analytic functions.