Multiplication and composition operators on the derivative Hardy space $S^{2}({\mathbb{D}})$

TL;DR

This paper introduces a new norm on the derivative Hardy space $S^{2}({ ext{D}})$, explores its properties including the reproducing kernel, and analyzes the behavior of composition and multiplication operators, revealing connections to 3-isometries and reducing subspaces.

Contribution

It defines an equivalent norm on $S^{2}({ ext{D}})$ with explicit kernel form, links it to 3-isometries, and characterizes operators and subspaces on this space.

Findings

The kernel of $S^{2}({ ext{D}})$ is a complete Nevanlinna-Pick kernel.

An upper bound for the norm of certain composition operators is established.

Multiplication operators that are $m$-isometries are fully characterized.

Abstract

In this paper we propose a different (and equivalent) norm on $S^{2} ({\mathbb{D}})$ which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of $S^{2}({\mathbb{D}})$ in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with $3$ -isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of $C_{\varphi}$ for a class of composition operators. We completely characterize multiplication operators which are $m$-isometries. As an application of the 3-isometry, we describe the reducing subspaces of $M_{\varphi}$ on $S^{2}({\mathbb{D}})$ when $\varphi$ is a finite Blaschke product of order 2.