Composition operators on some semi-Hilbert spaces

TL;DR

This paper characterizes various operator properties of composition and weighted composition operators on semi-Hilbert spaces induced by positive multiplication operators, extending the understanding of their structure and conditions for special classes.

Contribution

It provides new characterizations of composition operators on semi-Hilbert spaces using the framework of A-adjoint operators, including conditions for selfadjointness, normality, and isometry.

Findings

Characterization of A-selfadjoint, A-normal, A-quasinormal, and A-unitary composition operators.

Necessary and sufficient conditions for A-isometry and A-partial isometry.

Examples illustrating the main theoretical results.

Abstract

This paper investigates composition operators and weighted composition operators on semi-Hilbert spaces induced by positive multiplication operators on \( L^2(\mu) \). Within the framework of \( A \)-adjoint operators, we characterize conditions under which a composition operator \( C_\varphi \) is \( A \)-selfadjoint, \( A \)-normal, \( A \)-quasinormal, or \( A \)-unitary, where \( A = M_u \) denotes a positive multiplication operator. Additionally, we provide necessary and sufficient conditions for \( C_\varphi \) to be an \( A \)-isometry or an \( A \)-partial isometry. Analogous results are also obtained when \( A \) is a positive composition operator. Several examples are presented to illustrate the applicability of the main results.