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Localized Frames and Compactness

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Abstract

We introduce the concept of weak localization for continuous frames and use this concept to define a class of weakly localized operators. This class contains many important classes of operators, including: short time Fourier transform multipliers, Calderon–Toeplitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, some pseudodifferential operators, some Calderon–Zygmund operators, and many others. In this paper, we study the boundedness and compactness of weakly localized operators. In particular, we provide a characterization of compactness for weakly localized operators in terms of the behavior of their Berezin transforms.

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Acknowledgments

We would like to thank the two anonymous referees for their comments and suggestions. Research supported in part by National Science Foundation DMS Grant # 1101251.

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  1. Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC, 29634, USA

    Fawwaz Batayneh & Mishko Mitkovski

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  1. Fawwaz Batayneh
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  2. Mishko Mitkovski
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Correspondence to Mishko Mitkovski.

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Communicated by Karlheinz Gröchenig.

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Batayneh, F., Mitkovski, M. Localized Frames and Compactness. J Fourier Anal Appl 22, 568–590 (2016). https://doi.org/10.1007/s00041-015-9429-7

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