Composition-differentiation operators on $S^2(\mathbb{D})$

TL;DR

This paper characterizes boundedness, compactness, and spectra of composition-differentiation operators on the space $S^2$, providing explicit norm calculations and adjoint formulas for specific classes of inducing maps.

Contribution

It offers new characterizations and explicit computations for composition-differentiation operators on $S^2$, including norm, spectrum, and adjoint formulas for particular maps.

Findings

Characterization of bounded and compact operators on $S^2$

Explicit norm calculations for certain inducing maps

Spectrum identification and adjoint formulas for linear fractional maps

Abstract

We investigate composition-differentiation operators acting on the space $S^2$, the space of analytic functions on the open unit disk whose first derivative is in $H^2$. Specifically, we determine characterizations for bounded and compact composition-differentiation operators acting on $S^p$. In addition, for particular classes of inducing maps, we compute the norm, and identify the spectrum. Finally, for particular linear fractional inducing maps, we determine the adjoint of the composition-differentiation operator acting on weighted Bergman spaces which include $S^2, H^2$, and the Dirichlet space.