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 Suárez et al. (Integral Equ Oper Theory 75:197–233, 2013, Indiana Univ Math J 56(5):2185–2232, 2007) and limit operator techniques coming from similar problems on the sequence space $\ell ^p(\mathbb {Z})$ (Hagger et al. in J Math Anal Appl 437(1):255–291, 2016; Lindner and Seidel in J Funct Anal 267(3):901–917, 2014; Rabinovich et al. Integral Equ Oper Theory 30(4): 452–495, 1998 and references therein).