Adjoints of linear fractional composition operators on weighted Hardy spaces

TL;DR

This paper explores the structure of adjoint operators of linear fractional composition operators on various weighted Hardy spaces, extending known formulas from classical spaces to broader contexts.

Contribution

It generalizes the known adjoint formulas for linear fractional composition operators from classical Hardy and Bergman spaces to other weighted Hardy spaces.

Findings

Adjoint formulas hold exactly on classical spaces.

On other weighted Hardy spaces, formulas hold modulo compact operators.

Provides new insights into operator theory on weighted Hardy spaces.

Abstract

It is well known that on the Hardy space $H^2(\mathbb{D})$ or weighted Bergman space $A^2_{\alpha}(\mathbb{D})$ over the unit disk, the adjoint of a linear fractional composition operator equals the product of a composition operator and two Toeplitz operators. On $S^2(\mathbb{D})$, the space of analytic functions on the disk whose first derivatives belong to $H^2(\mathbb{D})$, Heller showed that a similar formula holds modulo the ideal of compact operators. In this paper we investigate what the situation is like on other weighted Hardy spaces.