Applications of Reproducing Kernels in composition operators

TL;DR

This paper demonstrates how reproducing kernel Hilbert space techniques can effectively analyze composition operators, providing new characterizations, classifications, and simplified proofs in the context of weighted Hardy spaces.

Contribution

It introduces novel reproducing kernel methods for classifying and analyzing composition operators, offering simpler proofs and a unifying framework.

Findings

Characterization of composition operators with adjoint as composition operators

Classification of bounded weighted composition operators

Simplified proofs of existing results using kernel techniques

Abstract

In this paper, we illustrate the effectiveness of reproducing kernel Hilbert space techniques in the study of composition operators. For weighted Hardy spaces on the unit disk, we characterize the composition operators whose adjoint is again a composition operator. Using reproducing kernel methods, we obtain a classification of bounded weighted composition operators acting between reproducing kernel Hilbert spaces. We also show that the reproducing kernel techniques yield simpler proofs of several known results, highlighting the role of reproducing kernels as a unifying structural tool in the analysis of composition operators.