Multiplication operators on $S^2(\mathbb D)$

TL;DR

This paper characterizes bounded, compact, and isometric multiplication operators on the space of analytic functions with derivatives in H^2, providing spectral and norm estimates and identifying isometries as constant modulus-one functions.

Contribution

It offers a complete characterization of multiplication operators on the space S^2, including boundedness, compactness, spectrum, and isometric conditions, which was not previously established.

Findings

Bounded and compact multiplication operators characterized.

Spectrum of multiplication operators determined.

Isometric multiplication operators are exactly those induced by unimodular constants.

Abstract

In this paper, we study the multiplication operators on $S^2$, the space of analytic functions on the open unit disk $\mathbb D$ whose first derivative is in $H^2$. Specifically, we characterize the bounded and the compact multiplication operators, establish estimates on the operator norm, and determine the spectrum. Finally, we prove that the isometric multiplication operators are precisely those induced by a constant function of modulus one.