Operator norm and numerical radius analogues of Cohen's inequality

Abstract

Let $D$ be an invertible multiplication operator on $L^2(X,\mu )$, and let $A$ be a bounded operator on $L^2(X,\mu )$. In this note we prove that $\Vert A{\Vert }^2\le \Vert DA\Vert \Vert D^{-1}A\Vert$, where $\Vert \cdot \Vert$ denotes the operator norm. If, in addition, the operators $A$ and $D$ are positive, we also have $w(A{)}^2\le w(DA)w(D^{-1}A)$, where $w$ denotes the numerical radius.

5 pages