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.

Full Text

Article Information

Title	Unbounded translation invariant operators on commutative hypergroups
Source	Methods Funct. Anal. Topology, Vol. 25 (2019), no. 3, 236-247
MathSciNet	MR4016211
Milestones	Received 03/10/2018
Copyright	The 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

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},
  MRNUMBER = {MR4016211},
       URL = {http://mfat.imath.kiev.ua/article/?id=1209},
}

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