Limit operators, compactness and essential spectra on bounded symmetric domains

Abstract

This paper is a follow-up to a recent article about the essential spectrum of Toeplitz operators acting on the Bergman space over the unit ball. As mentioned in the said article, some of the arguments can be carried over to the case of bounded symmetric domains and some cannot. The aim of this paper is to close the gaps to obtain comparable results for general bounded symmetric domains. In particular, we show that a Toeplitz operator on the Bergman space A p ν is Fredholm if and only if all of its limit operators are invertible. Even more generally, we show that this is in fact true for all band-dominated operators, an algebra that contains the Toeplitz algebra. Moreover, we characterize compactness and explain how the Berezin transform comes into play. In particular, we show that a bounded linear operator is compact if and only if it is band-dominated and its Berezin transform vanishes at the boundary. For p = 2 "band-dominated" can be replaced by "contained in the Toeplitz algebra".

Limit operators techniques on general metric measure spaces of bounded geometry

“…For every x ∈ X we can choose biholomorphic maps φ x : X → X with φ x (0) = x and x → φ x (y) continuous for all y ∈ X ([18, Lemma 5]). Moreover, the standard transformation properties of µ show that µ • φ x ≪ µ ≪ µ • φ x and we can choose where g denotes the genus of X (see again [11,18]). As h(y, x) is actually a polynomial in y and x, and has no zeroes in X × X, x → h x (y) is continuous for all y ∈ X as well.…”

“…In the last few years, limit operator techniques have been applied to a variety of situations such as operators on Fock spaces [14], on Bergman spaces [17,18], and also on uniformly discrete metric measure spaces of bounded geometry [36,40]. Further applications to spectral theory for operator families can be found in [2,3].…”

“…P f ](z) = X f (y)h(z, y) −ν−g dµ(y),and the standard transformation properties of h and µ (see again[11,18]), we arrive at[U p x P (U p x ) −1 f ](z) = h(z, x) (ν+g)(1− 2 p ) X f (y)h(y, x) −(ν+g)(1− 2 p ) h(z, y) −ν−g dµ(y).…”

Berezin transform and Toeplitz operators on polygonal domains

“…7.5.]. The Berezin transform can also be used to characterize the Fredholm properties or the essential spectrum of T g , see the recent paper [3] and the references therein.…”

Toeplitz Operators on Pluriharmonic Function Spaces: Deformation Quantization and Spectral Theory

“…If we again assume Ω to be C n or a bounded symmetric domain, the essential spectrum is well understood: It consists of the boundary values of its symbols (in a certain sense), c.f. [1,17,18,20] and references therein for the most recent results.…”