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

TL;DR

This paper analyzes the spectral properties and norms of weighted composition and differentiation operators on Hardy and weighted Bergman spaces, providing explicit formulas and estimates under specific conditions.

Contribution

It offers new spectral descriptions and norm estimates for combined weighted composition and differentiation operators on Hardy and Bergman spaces, including explicit norm calculations for certain cases.

Findings

Spectrum of the sum of operators characterized when fixed point is inside the unit disk.

Upper and lower bounds for the operator norm on Hardy space established.

Exact norm of a specific composition-differentiation operator with linear symbol determined.

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 $|b|<1$.