Localization operators and exponential weights for modulation spaces
The authors extend the study of localization operators in [\textit{E.\,Cordero} and \textit{K.\,Gröchenig}, J.~Funct.\ Anal.\ 205, No.\,1, 107--131 (2003; Zbl 1047.47038)] by considering the case of their window functions belonging to the Gelfand--Shilov space \(S^{(1)}\) of analytic functions, and the symbols from its strong dual, the space of tempered ultra-distributions of Beurling type. Necessary and sufficient conditions for the localization operators to be bounded or to belong to a Schatten class are given. The results are formulated in terms of modulation spaces with weights which may have exponential growth. In particular, the authors study symbols defined by ultra-distributions with compact support, that give trace class localization operators.
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