Order convolution and vector-valued multipliers

U. B. Tewari¹

¹ Department of Mathematics and Statistics Indian Institute of Technology Kanpur Kanpur 208 016, India

Colloquium Mathematicum, Tome 108 (2007) no. 1, pp. 53-61.

Résumé

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 $L^1(I,Y)$.

Détail

DOI : 10.4064/cm108-1-5

Keywords: infty usual topology define max becomes locally compact commutative topological semigroup banach space lebesgue integrable functions becomes commutative semisimple banach algebra order convolution multiplication bounded linear operator called multiplier star star space multipliers determined johnson lahr banach space banach space x valued bochner integrable functions becomes banach module suppose banach spaces bounded linear operator called multiplier star star paper characterize multipliers