Adjoints of composition operators with rational symbol

TL;DR

This paper derives a concrete formula for the adjoint of composition operators with rational symbols on the Hardy space, expanding understanding of their structure and providing explicit calculations for specific cases.

Contribution

It introduces a new explicit formula for the adjoint of composition operators with rational symbols, building on prior techniques and enabling detailed analysis of these operators.

Findings

Derived a concrete formula for the adjoint operator

Compared new formula with existing results in specific examples

Enhanced understanding of composition operators with rational symbols

Abstract

Building on techniques developed by Cowen and Gallardo-Guti\'{e}rrez, we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the Hardy space $H^{2}$. We consider some specific examples, comparing our formula with several results that were previously known.