Weighted Composition--Differentiation Operator on the Hardy and Weighted Bergman Spaces

Abstract: In this paper, we consider the sum of weighted composition operator $C_{\psi_{0},\varphi_{0}}$ and the weighted composition--differentiation operator $D_{\psi_{n},\varphi_{n},n}$ on the Hardy and weighted Bergman spaces. We describe the spectrum of a compact operator $C_{\psi_{0},\varphi_{0}}+D_{\psi_{n},\varphi_{n},n}$ when the fixed point $w$ of $\varphi_{0}$ and $\varphi_{n}$ is inside the open unit disk and $\psi_{n}$ has a zero at $w$ of order at least $n$. Also the lower estimate and the upper estimate on the norm of a weighted composition--differentiation operator on the Hardy space $H^{2}$ are obtained. Furthermore, we determine the norm of a composition--differentiation operator $D_{\varphi,n}$, acting on the Hardy space $H^{2}$, in the case where $\varphi(z)=bz$ for some complex number $b$ that $|b|<1$.