Complex Symmetric Composition Operators on $H^2$

TL;DR

This paper identifies complex symmetric composition operators on the Hardy space with linear-fractional symbols that are not automorphisms, addressing open questions in operator theory.

Contribution

It demonstrates the existence of such operators and provides partial answers to longstanding open problems in the field.

Findings

Existence of complex symmetric composition operators with non-automorphic symbols.

Answers to Noor's recent question about these operators.

Partial resolution of Garcia and Hammond's original problem.

Abstract

In this paper, we find complex symmetric composition operators on the classical Hardy space whose symbols are linear-fractional but not automorphic. In doing so, we answer a recent question of Noor, and partially answer the original problem posed by Garcia and Hammond.