On operators which commute with translations and convolutions

Document Type : Original Article

Author

Department of Mathematics, Semnan University, Semnan, Iran

Abstract

Let $G$ be a locally compact group. In this paper, we study bounded linear operators on subspaces of $L^\infty(G)$ which commutes with translation and convolution operators. We prove, among the other things, that $G$ is compact if and only if for every bounded linear operator $T:L^\infty(G)\to L^\infty(G)$, $\phi T(f)=T(\phi f)$ $(\phi\in L^1(G)$ and $f\in L^\infty(G))$ implies that $FT(f)=T(Ff)$ for all $F\in L^\infty(G)^*$ and $f\in L^1(G)$.  To every $f\in L^\infty(G)$, we associate the operator $T_f:L^1(G)\to L^\infty(G)$ defined by $T_f(\phi)=f\phi$. We can embed $L^\infty(G)$ into $\mathcal B(L^1(G),L^\infty(G))$. $L^\infty(G)$ is a subspace of $\mathcal B(L^1(G),L^\infty(G))$ with respect to the strong operator topology. Let $T\in \mathcal B(L^\infty(G))$ be continuous with respect to the strong operator topology on $L^\infty(G)$. We show that $T$ commutes with translations if and only $T$ commutes with convolutions.

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References

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Measure Algebras and Applications
Volume 2, Issue 2 - Serial Number 3
December 2024
Pages 141-150

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History

  • Receive Date: 05 September 2024
  • Revise Date: 21 November 2024
  • Accept Date: 15 December 2024

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