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Abstract

Let $I=(0,\infty )$ with the usual topology. For $x,y\in I$, we define $xy= \max(x,y)$. Then $I$ becomes a locally compact commutative topological semigroup. The Banach space $L^1(I)$ of all Lebesgue integrable functions on $I$ becomes a commutative semisimple Banach algebra with order convolution as multiplication. A bounded linear operator $T$ on $L^1(I)$ is called a multiplier of $L^1(I)$ if $T(f\star g)=f\star Tg$ for all $f,g \in L^1(I)$. The space of multipliers of $L^1(I)$ was determined by Johnson and Lahr. Let $X$ be a Banach space and $L^1(I,X)$ be the Banach space of all $X$-valued Bochner integrable functions on $I$. We show that $L^1(I,X)$ becomes an $L^1(I)$-Banach module. Suppose $X$ and $Y$ are Banach spaces. A bounded linear operator $T$ from $L^1(I,X)$ to $L^1(I,Y)$ is called a multiplier if $T(f\star g)=f\star Tg$ for all $f\in L^1(I)$ and $g\in L^1(I,X)$. In this paper, we characterize the multipliers from $L^1(I,X)$ to $L^1(I,Y)$.