The Essential Spectrum of Toeplitz Operators on the Unit Ball

Abstract: In this paper we study the Fredholm properties of Toeplitz operators acting on 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 unit ball. 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 $T_f$ is given by the boundary values of $f$. We extend this result to all operators in the algebra generated by Toeplitz operators with bounded symbol (in a sense to be made precise down below). The main ideas are based on the work of Suarez et al. and limit operator techniques coming from similar problems on the sequence space $\ell^p(\mathbb{Z})$.