Geometric mean and norm Schwarz inequality

Tsuyoshi Ando

doi:10.1215/20088752-3158073

February 2016 Geometric mean and norm Schwarz inequality

Tsuyoshi Ando

Ann. Funct. Anal. 7(1): 1-8 (February 2016). DOI: 10.1215/20088752-3158073

Abstract

Positivity of a $2\times2$ operator matrix $[\begin{smallmatrix}A&B\\B^{*}&C\end{smallmatrix}]\geq0$ implies $\sqrt{\|A\|\cdot\|C\|}\geq\|B\|$ for operator norm $\|\cdot\|$ . This can be considered as an operator version of the Schwarz inequality. In this situation, for $A,C\geq0$ , there is a natural notion of geometric mean $A\sharp C$ , for which $\sqrt{\|A\|\cdot\|C\|}\geq\|A\sharp C\|$ . In this paper, we study under what conditions on $A$ , $B$ , and $C$ or on $B$ alone the norm inequality $\sqrt{\|A\|\cdot\|C\|}\geq\|B\|$ can be improved as $\|A\sharp C\|\geq\|B\|$ .

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Citation Download Citation

Tsuyoshi Ando "Geometric mean and norm Schwarz inequality," Annals of Functional Analysis 7(1), 1-8, (February 2016). https://doi.org/10.1215/20088752-3158073

Received: 5 December 2014; Accepted: 17 December 2014; Published: February 2016

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