Toeplitz operators with BMO symbols on the Segal-Bargmann space

Abstract

We show that Zorboska’s criterion for compactness of Toeplitz operators with BMO 1 \text {BMO}^1 symbols on the Bergman space of the unit disc holds, by a different proof, for the Segal-Bargmann space of Gaussian square-integrable entire functions on C n \mathbb {C}^n . We establish some basic properties of BMO p \text {BMO}^p for p ≥ 1 p \geq 1 and complete the characterization of bounded and compact Toeplitz operators with BMO 1 \text {BMO}^1 symbols. Via the Bargmann isometry and results of Lo and Englis̆, we also give a compactness criterion for the Gabor-Daubechies “windowed Fourier localization operators” on L 2 ( R n , d v ) L^2(\mathbb {R}^n, dv) when the symbol is in a BMO 1 \text {BMO}^1 Sobolev-type space. Finally, we discuss examples of the compactness criterion and counterexamples to the unrestricted application of this criterion for the compactness of Toeplitz operators.