2017/05/12 by Hagger, Raffael · 1 citation
#47A10 #47A53 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary: 47B35 #Secondary: 32A36
paper · doi:10.48550/arxiv.1705.04553
In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces Apν(\mathbbBn), where p ∈ (1,∞) and \mathbbBn ⊂ ℂn denotes the n-dimensional open unit ball. Let f be a continuous function on the Euclidean closure of \mathbbBn. It is well-known that then the corresponding Toeplitz operator Tf is Fredholm if and only if f has no zeros on the boundary ∂\mathbbBn. 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 Suarez et al. and limit operator techniques coming from similar problems on the sequence space ℓp(ℤ).