"Tauberian theorems for operators and functions"Luef, FranzWe present compactness results about localization operators and quantization schemes that are deduced from Wiener-Tauberian theorems for the convolution of functions with trace class operators and the convolution of two trace class operators in the sense of quantum harmonic analysis. For rank-one operators turn these convolutions into well-known objects in time-frequency analysis: localization operators and spectrograms. As a consequence we show that a compactness result by Fernandez-Galbis about localization operators is equivalent to Wiener's Tauberian Theorem for functions. Furthermore we discuss an extension of compactness results for Toeplitz operators on Bargmann-Fock spaces in terms of the Berezin transform due to Bauer-Isralowitz to a more general class of localization operators. This is based on joint work with Eirik Skrettingland. |
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