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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.

### Article Information

 Title Unbounded translation invariant operators on commutative hypergroups Source Methods Funct. Anal. Topology, Vol. 25 (2019), no. 3, 236-247 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},
URL = {http://mfat.imath.kiev.ua/article/?id=1209},
}

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