We present localization operators via the short-time Fourier transform. For both modulation and ultra-modulation spaces framework, well-known results about boundedness and Schatten-von Neumann class are reported. Asymptotic eigenvalues’ distribution and decay and smoothness properties for L2- eigenfunctions are exhibited. Eventually, we make a conjecture about smoothness of L2-eigenfunctions for localization operators with Gelfand–Shilov windows and symbols in ultra-modulation spaces.
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