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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 $\|A\|^2\le \|D A\| \, \|D^{-1} A\|$, where $\|\cdot\|$ denotes the operator norm. If,in addition, the operators $A$ and $D$ are positive, we also have $w(A)^2 \lew(D A) \, w(D^{-1} A)$, where $w$ denotes the numerical radius.

Operator norm and numerical radius analogues of Cohen's inequality