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arXiv · 2108.05645

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

Mahsa Fatehi
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Abstract

In this paper, we consider the sum of weighted composition operator $C_{ψ_{0},φ_{0}}$ and the weighted composition--differentiation operator $D_{ψ_{n},φ_{n},n}$ on the Hardy and weighted Bergman spaces. We describe the spectrum of a compact operator $C_{ψ_{0},φ_{0}}+D_{ψ_{n},φ_{n},n}$ when the fixed point $w$ of $φ_{0}$ and $φ_{n}$ is inside the open unit disk and $ψ_{n}$ 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 $D_{φ,n}$, acting on the Hardy space $H^{2}$, in the case where $φ(z)=bz$ for some complex number $b$ that $|b|<1$.

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Mahsa Fatehi. 2021-08-12. Weighted Composition--Differentiation Operator on the Hardy and Weighted Bergman Spaces. https://arxiv.org/abs/2108.05645

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