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

Abstract

In this paper, we consider the sum of weighted composition operator $C_{{\psi }_0,{\phi }_0}$ and the weighted composition--differentiation operator $D_{{\psi }_n,{\phi }_n,n}$ on the Hardy and weighted Bergman spaces. We describe the spectrum of a compact operator $C_{{\psi }_0,{\phi }_0}+D_{{\psi }_n,{\phi }_n,n}$ when the fixed point $w$ of ${\phi }_0$ and ${\phi }_n$ 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 $D_{\phi ,n}$, acting on the Hardy space $H^2$, in the case where $\phi (z)=bz$ for some complex number $b$ that $|b|<1$.