On $(p,q)-$centralizers of certain Banach algebras

Abstract

Let $A$ be an algebra with a right identity. In this paper, we study $(p,q)-$centralizers of $A$ and show that every $(p,q)-$centralizer of $A$ is a two-sided centralizer. In the case where, $A$ is normed algebra, we also prove that $(p,q)-$centralizers of $A$ are bounded. Then, we apply the results for some group algebras and verify that $L^1(G)$ has a nonzero weakly compact $(p,q)-$centralizer if and only if $G$ is compact and the center of $L^1(G)$ is non-zero. Finally, we investigate $(p,q)-$Jordan centralizers of $A$ and determine them.