Generalized Weighted Composition Operator on Weighted Hardy Spaces

Abstract

Let $\varphi$ be an analytic self-map of the open unit disc $\mathbb {D}$ in the finite complex plane $\mathbb{C}$ and $\psi$ be an analytic map of the open unit disc $\mathbb {D}$ to $\mathbb{C}$. Let $ C_{\varphi}\;,M_{\psi}\;and\;D^n $ be the composition, multiplication and differentiation operators defined by $C_{\varphi}f=f\circ\varphi,\; M_{\psi}f=\psi.f \;and\; D^{n}f= f^{n} $ respectively. In this paper, we shall study the boundedness and compactness of the generalized weighted composition operator $W_{\psi\;\varphi}D^{n}$ defined by $W_{\psi\;\varphi}D^{n}f\;=\;\psi.(f^{n}\circ\phi)$ on weighted Hardy spaces by using the orthonormal basis of the weighted Hardy spaces $H^{2}(\beta)$.