Adjoints of generalized composition operators with rational symbol

Abstract

Given $\varphi:\mathbb{U}\rightarrow\mathbb{U}$, an analytic self-map of the open unit disc in complex plane, the composition operator $C_{\varphi}$ is defined by $C_{\varphi}f=f\circ\varphi$ for $f$ belonging to some Hilbert space of analytic functions on $\mathbb{U}$. In the present paper, we introduce a generalization of the composition operators and reproducing kernel functions on the weighted Hardy spaces. We also obtain the adjoints of generalized composition operators with rational symbol acting on the Hardy, Bergman and Dirichlet spaces.