The Essential Spectrum of Toeplitz Operators on the Unit Ball

摘要

In this paper we study the Fredholm properties of Toeplitz operators actingon weighted Bergman spaces $A^p_{\nu}(\mathbb{B}^n)$, where $p \in (1,\infty)$and $\mathbb{B}^n \subset \mathbb{C}^n$ denotes the $n$-dimensional open unitball. Let $f$ be a continuous function on the Euclidean closure of$\mathbb{B}^n$. It is well-known that then the corresponding Toeplitz operator$T_f$ is Fredholm if and only if $f$ has no zeros on the boundary$\partial\mathbb{B}^n$. As a consequence, the essential spectrum of $T_f$ isgiven by the boundary values of $f$. We extend this result to all operators inthe algebra generated by Toeplitz operators with bounded symbol (in a sense tobe made precise down below). The main ideas are based on the work of Suarez etal. and limit operator techniques coming from similar problems on the sequencespace $\ell^p(\mathbb{Z})$.