Composition-differentiation operators on the Dirichlet space

TL;DR

This paper studies composition-differentiation operators on the Dirichlet space, providing characterizations of their boundedness, compactness, and Hilbert-Schmidt properties, along with formulas for their adjoints, norms, and spectra.

Contribution

It offers new characterizations and explicit formulas for these operators on the Dirichlet space, advancing understanding of their spectral and norm properties.

Findings

Characterized bounded, compact, and Hilbert-Schmidt composition-differentiation operators.

Derived adjoint formulas for specific classes of inducing maps.

Computed operator norms and identified spectra.

Abstract

We investigate composition-differentiation operators acting on the Dirichlet space of the unit disk. Specifically, we determine characterizations for bounded, compact, and Hilbert-Schmidt composition-differentiation operators. In addition, for particular classes of inducing maps, we derive an adjoint formula, compute the norm, and identify the spectrum.