Abstract/Summary

In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces Ap ν(Bn), where p ∈ (1, ∞) and Bn ⊂ Cn denotes the n-dimensional open unit ball. Let f be a continuous function on the Euclidean closure of Bn. It is well-known that then the corresponding Toeplitz operator Tf is Fredholm if and only if f has no zeros on the boundary ∂Bn. As a consequence, the essential spectrum of Tf 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 Su´arez 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 p(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).