Composition operators on some semi-Hilbert spaces

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.