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Translation Preserving Operators on Locally Compact Abelian Groups

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Abstract

We study translation preserving operators, that is operators commuting with translations by a closed subgroup of a locally compact abelian group. We show that there is a one-to-one correspondence between these operators and range operators. Furthermore, we obtain a necessary condition for a translation preserving operator to be Hilbert–Schmidt or of finite trace in terms of its range operator. The last result is proved in the case that the subgroup is discrete or compact.

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Acknowledgements

We are indebted to Dr. R. Fa’al for a significant contribution on the proof of part \((1 \Rightarrow 2)\) of Theorem 2.2. We are also indebted to Professor Hartmut Fuhr for valuable comments and remarks on the proof of Theorem 2.2. We thank Professor Kenneth A. Ross for stimulating discussions on an earlier version of this paper. We are also grateful to the referee for constructive comments and fruitful suggestions. The second author is supported by “Iran National Science Foundation INSF”.

Funding

This article was funded by Ferdowsi University of Mashhad. The second author is supported by “Iran National Science Foundation INSF”.

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Authors and Affiliations

  1. Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran

    M. Mortazavizadeh, R. Raisi Tousi & R. A. Kamyabi Gol

  2. Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

    R. Raisi Tousi

  3. Center of Excellence in Analysis on Algebraic Structures (CEAAS), Mashhad, Iran

    R. A. Kamyabi Gol

Authors
  1. M. Mortazavizadeh
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  2. R. Raisi Tousi
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  3. R. A. Kamyabi Gol
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Correspondence to R. Raisi Tousi.

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Mortazavizadeh, M., Tousi, R.R. & Gol, R.A.K. Translation Preserving Operators on Locally Compact Abelian Groups. Mediterr. J. Math. 17, 126 (2020). https://doi.org/10.1007/s00009-020-01562-y

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  • DOI: https://doi.org/10.1007/s00009-020-01562-y

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