The Essential Spectrum of Toeplitz Operators on the Unit Ball

In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces A ν p ( B n ) , where p ∈ ( 1 , ∞ ) and B n ⊂ C n denotes the n -dimensional open unit ball. Let f be a continuous function on the Euclidean closure of 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 ∂ 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 ℓ 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).