Convolutions for Localization Operators
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The theory of quantum harmonic analysis on phase space introduced by Werner is presented and formulated precisely using the terminology of time-frequency analysis and abstract harmonic analysis. Convolutions of functions with operators and of operators with operators are introduced, along with a corresponding Fourier transform of operators the Fourier-Wigner transform. Using these concepts we formulate and prove a version of Wiener s Tauberian theorem for operators due to Werner. The main novel result of the thesis is a formulation of the so- called localization operators using the convolution of a function with an operator, which gives a conceptual framework for localization operators and an extension of results by Bayer and Gröchenig. The connection to quantum harmonic analysis provides new perspectives on results in time-frequency analysis. In particular, Lieb s uncertainty principle is seen to be a special case of a Hausdorff-Young inequality for operators, which in turn leads to an improvement of this Hausdorff-Young inequality. We also show a generalization of the Berezin-Lieb inequalities, and relate this and the convolutions to results by Klauder and Skagerstam. The theory of Banach modules is used to prove new results on the convolutions, and the Fourier-Wigner transform is shown to be related to the so-called Arveson spectrum. Finally the convolutions are considered in the context of modulation spaces, inspired by the existing literature on localization operators and modulation spaces.