Toeplitz operators with bmo symbols on the segal-bargmann space

We show that Zorboska's criterion for compactness of Toeplitz operators with BMO¹ 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 ℂⁿ. We establish some basic properties of BMO^p for p ≤ 1 and complete the characterization of bounded and compact Toeplitz operators with BMO¹ symbols. Via the Bargmann isometry and results of Lo and Engliš, we also give a compactness criterion for the Gabor-Daubechies "windowed Fourier localization operators" on L²(ℝⁿ, dv) when the symbol is in a BMO1 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.