Abstract
Time–frequency localization operators are a quantization procedure that maps symbols on \({\mathbb{R}^{2d}}\) to operators and depends on two window functions. We study the range of this quantization and its dependence on the window functions. If the If the short-time Fourier transform of the windows does not have any zero, then the range is dense in the Schatten p-classes. The main tool is new version of the Berezin transform associated to operators on \({L^{2}(\mathbb{R}^{d})}\). Although some results are analogous to results about Toeplitz operators on spaces of holomorphic functions, the absence of a complex structure requires the development of new methods that are based on time-frequency analysis.
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K. G. was supported in part by the Project P26273-N25 of the Austrian Science Fund (FWF).
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Bayer, D., Gröchenig, K. Time–Frequency Localization Operators and a Berezin Transform. Integr. Equ. Oper. Theory 82, 95–117 (2015). https://doi.org/10.1007/s00020-014-2208-z
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DOI: https://doi.org/10.1007/s00020-014-2208-z