Composition-differentiation operators acting on certain Hilbert spaces of analytic functions

Year 2024, Volume: 53 Issue: 3, 586 - 594, 27.06.2024

Yazdan Bayat , Ali Abkar

https://doi.org/10.15672/hujms.1241783

Abstract

We study composition-differentiation operators acting on the Bergman and Dirichlet space of the open unit disk. We first characterize the compactness of composition-differentiation operator on weighted Bergman spaces. We shall then prove that for an analytic self-map $\varphi$ on the open unit disk $\mathbb{D}$, the induced composition-differentiation operator is bounded with dense range if and only if $\varphi$ is univalent and the polynomials are dense in the Bergman space on $\Omega:=\varphi(\mathbb{D})$.

Keywords

composition-differentiation operator, Bergman space, Dirichlet space, compact operator, dense-range operator

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