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Unbounded translation invariant operators on commutative hypergroups


Abstract

Let $K$ be a commutative hypergroup. In this article, we study the unbounded translation invariant operators on $L^p(K),\, 1\leq p \leq \infty.$ For $p \in \{1,2\},$ we characterize translation invariant operators on $L^p(K)$ in terms of the Fourier transform. We prove an interpolation theorem for translation invariant operators on $L^p(K)$ and we also discuss the uniqueness of the closed extension of such an operator on $L^p(K)$. Finally, for $p \in \{1,2\},$ we prove that the space of all closed translation invariant operators on $L^p(K)$ forms a commutative algebra over the field of complex numbers. We also prove Wendel's theorem for densely defined closed linear operators on $L^1(K).$

Key words: Unbounded multipliers, translation invariant operators, unbounded operators, hypergroups, Fourier transform.


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Article Information

TitleUnbounded translation invariant operators on commutative hypergroups
SourceMethods Funct. Anal. Topology, Vol. 25 (2019), no. 3, 236-247
MilestonesReceived 03/10/2018
CopyrightThe Author(s) 2019 (CC BY-SA)

Authors Information

Vishvesh Kumar
Department of Mathematics, Indian Institute of Technology, Delhi, Delhi - 110 016, India

N. Shravan Kumar
Department of Mathematics, Indian Institute of Technology, Delhi, Delhi - 110 016, India

Ritumoni Sarma
Department of Mathematics, Indian Institute of Technology, Delhi, Delhi - 110 016, India


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Citation Example

Vishvesh Kumar, N. Shravan Kumar, and Ritumoni Sarma, Unbounded translation invariant operators on commutative hypergroups, Methods Funct. Anal. Topology 25 (2019), no. 3, 236-247.


BibTex

@article {MFAT1209,
    AUTHOR = {Vishvesh Kumar and N. Shravan Kumar and Ritumoni Sarma},
     TITLE = {Unbounded translation invariant operators on commutative hypergroups},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {25},
      YEAR = {2019},
    NUMBER = {3},
     PAGES = {236-247},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=1209},
}


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