Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators

Abstract

We study mixed-state localization operators from the perspective of Werner’s operator convolutions which allows us to extend known results from the rank-one case to trace class operators. The idea of localizing a signal to a domain in phase space is approached from various directions such as bounds on the spreading function, probability densities associated to mixed-state localization operators, positive operator valued measures, positive correspondence rules and variants of Tauberian theorems for operator translates. Our results include a rigorous treatment of multiwindow-STFT filters and a characterization of mixed-state localization operators as positive correspondence rules. Furthermore we provide a description of the Cohen class in terms of Werner’s convolution of operators and deduce consequences on positive Cohen class distributions, an uncertainty principle, uniqueness and phase retrieval for general elements of Cohen’s class.

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Notes

  1. 1.

    This follows from Theorem 1 on p. 79 of [24], as \({\mathcal {T}}^1\) is the dual space of the compact operators.

  2. 2.

    We have ignored one issue: we need to restrict \(\Omega \) to bounded subsets to ensure that \(\chi _{\Omega }\in L^1({\mathbb {R}}^{2d})\). The technical details needed to circumvent this issue are given in the proof of Lemma 3.1 in [46].

  3. 3.

    By “up to sets of Lebesgue measure zero” we mean that we regard two sets \(\Omega ,\Omega ^\prime \) to be equal if \(\mu (\Omega \triangle \Omega ^{\prime })=0\), where \(\triangle \) is the symmetric difference of sets.

  4. 4.

    In the sense that \(Q(\pi (z)\psi )=T_z(Q(\psi ))\) for \(z\in {\mathbb {R}}^{2d}\) and \(\psi \in L^2({\mathbb {R}}^d)\).

  5. 5.

    The proposition requires that \(\chi _{\Omega }\star 2^dP=L_{\chi _{\Omega }}\) is compact. Even though P is not a compact operator, the operator \(L_{\chi _{\Omega }}\) is compact whenever \(\mu (\Omega )<\infty \), since \(\chi _{\Omega }\in L^2({\mathbb {R}}^{2d})\) in this case and \(L_f\in {\mathcal {T}}^2\) whenever \(f\in L^2({\mathbb {R}}^{2d})\) by Pool’s theorem [60].

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Correspondence to Franz Luef.

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Luef, F., Skrettingland, E. Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators. J Fourier Anal Appl 25, 2064–2108 (2019). https://doi.org/10.1007/s00041-019-09663-3

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Keywords

  • Localization operators
  • Cohen class
  • Uncertainty principle
  • Phase retrieval
  • Positive operator valued measures

Mathematics Subject Classification

  • 47G30
  • 35S05
  • 46E35
  • 47B10