فهرست نشریات علمی دسترسی آزاد ایران

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.