Positivity of a operator matrix implies for operator norm . This can be considered as an operator version of the Schwarz inequality. In this situation, for , there is a natural notion of geometric mean , for which . In this paper, we study under what conditions on , , and or on alone the norm inequality can be improved as .
"Geometric mean and norm Schwarz inequality." Ann. Funct. Anal. 7 (1) 1 - 8, February 2016. https://doi.org/10.1215/20088752-3158073