On the Pseudospectra of Berezin-Toepolitz Operators

Abstract

The purpose of this note is to show that microlocal techniques can also be applied to the study of the pseudospectra of matrices (1.2) such as that discussed in a recent paper by Trefethen and Chapman [18] (and generalizations). In this light we interpret the twist condition as Hormander's solvability condition on the Poisson bracket of the real and imaginary parts of the symbol of a pseudodifferential operator. The connection between Hormander's condition and pseudospectra was first made by M. Zworski in [21]. In this paper we construct pseudomodes for Berezin-Toeplitz operators under condition (1.4) on the (smooth) symbol. Although we will discuss our results in detail in the next section, we should mention some limitations of our work. The methods of Trefethen and Chapman apply to rough symbols, f, and they obtain exponentially small error terms. For analytic symbols, it is very likely that exponentially small estimates (in the Toeplitz setting) can be achieved by microlocal methods, as has been done in [10] for pseudodifferential operators. The problem of dealing with general non-smooth symbols, however, is much more challenging. Trefethen and Chapman's main theorem includes a global condition on the symbol (in addition to 1.2), and they present compelling numerical evidence that global conditions on non-smooth symbols are necessary for the existence of "good" pseudomodes [18]. This is a very interesting issue that we do not address here. On the other hand, our results for smooth symbols are fairly general and include a number of cases not covered by the results in [18] (e. g. the "Scottish flag" matrix). Furthermore, the pseudomodes we construct are localized in phase space, sharpening the localization results of [18].