Geometric mean and norm Schwarz inequality

Abstract

Positivity of a $2\times 2$ operator matrix implies $\sqrt{\Vert A\Vert \cdot \Vert C\Vert }\ge \Vert B\Vert$ for operator norm $\Vert \cdot \Vert$. This can be considered as an operator version of the Schwarz inequality. In this situation, for $A,C\ge 0$, there is a natural notion of geometric mean $A♯C$, for which $\sqrt{\Vert A\Vert \cdot \Vert C\Vert }\ge \Vert A♯C\Vert$. In this paper, we study under what conditions on $A$, $B$, and $C$ or on $B$ alone the norm inequality $\sqrt{\Vert A\Vert \cdot \Vert C\Vert }\ge \Vert B\Vert$ can be improved as $\Vert A♯C\Vert \ge \Vert B\Vert$.