Spectral theory of multiplication operators on Hardy-Sobolev spaces

TL;DR

This paper investigates the spectral characteristics of multiplication operators on Hardy-Sobolev spaces, including spectrum, essential spectrum, and Fredholm properties, providing a comprehensive analysis of their functional behavior.

Contribution

It introduces a detailed spectral and Fredholm theory for multiplication operators on Hardy-Sobolev spaces, extending existing operator theory in this context.

Findings

Computed the spectrum of multiplication operators on Hardy-Sobolev spaces

Determined the essential spectrum of these operators

Developed a Fredholm theory for multiplication operators

Abstract

For a pointwise multiplier $\varphi$ of the Hardy-Sobolev space $H^2_\beta$ on the open unit ball $\bn$ in $\cn$, we study spectral properties of the multiplication operator $M_\varphi: H^2_\beta\to H^2_\beta$. In particular, we compute the spectrum and essential spectrum of $M_\varphi$ and develop the Fredholm theory for these operators.