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Applications of Reproducing Kernels in composition operators

Preeti Kumari, P. Muthukumar, Antti Rasila

TL;DR

This work develops a unified RKHS perspective on composition and weighted composition operators, showing how reproducing kernels streamline the analysis across classical one-variable Hardy spaces and their weighted variants, as well as multivariable domains. Key results include an explicit norm formula and non-attainment for inner-symbol composition operators, a complete kernel-based characterization of when adjoints are themselves composition operators, and a general RKHS framework for weighted composition operators with precise boundedness criteria. The methodology yields simpler proofs of known results and extends to several complex variables, enhancing structural understanding of operator behavior via kernel techniques. The findings have potential implications for spectral properties and norm estimates in RKHS settings beyond the classical Hardy spaces.

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.

Applications of Reproducing Kernels in composition operators

TL;DR

This work develops a unified RKHS perspective on composition and weighted composition operators, showing how reproducing kernels streamline the analysis across classical one-variable Hardy spaces and their weighted variants, as well as multivariable domains. Key results include an explicit norm formula and non-attainment for inner-symbol composition operators, a complete kernel-based characterization of when adjoints are themselves composition operators, and a general RKHS framework for weighted composition operators with precise boundedness criteria. The methodology yields simpler proofs of known results and extends to several complex variables, enhancing structural understanding of operator behavior via kernel techniques. The findings have potential implications for spectral properties and norm estimates in RKHS settings beyond the classical Hardy spaces.

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.
Paper Structure (5 sections, 16 theorems, 97 equations)

This paper contains 5 sections, 16 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Composition operators induced by inner functions
  3. Adjoint of the composition operators
  4. Weighted composition operators
  5. Composition operators in several variables

Key Result

Theorem 2.1

Let ${\varphi}$ be an inner function with ${\varphi}(0)\neq 0$ and $f\in H^2({\mathbb D})$. Then if and only if $f\equiv 0$. That is, $\|C_{\varphi}\|$ is never attained by any function in $H^2({\mathbb D})$.

Theorems & Definitions (29)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 19 more