Spectral theory of multiplication operators on Hardy-Sobolev spaces

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.