# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Cauchy-Schwarz and means inequalities for elementary operators into norm idealsHTML articles powered by AMS MathViewer

by Danko R. Jocić
Proc. Amer. Math. Soc. 126 (1998), 2705-2711 Request permission

## Abstract:

The Cauchy-Schwarz norm inequality for normal elementary operators $\left \Vvert \sum _{n=1}^\infty A_nXB_n \right \Vvert \leq \left \Vvert (\sum _{n=1}^\infty A_n^*A_n)^{1/2}X (\sum _{n=1}^\infty B_n^*B_n)^{1/2} \right \Vvert ,$ implies a means inequality for generalized normal derivations $\left \Vvert \frac {AX+XB}2 \right \Vvert \leq \Vvert X \Vvert ^{1-\frac 1r} \left \Vvert \frac {|A|^rX+X|B|^r}2 \right \Vvert ^\frac 1r,$ for all $r\ge 2$, as well as an inequality for normal contractions $A$ and $B$ $\left \Vvert (I-A^*A) ^\frac 12X(I-B^*B)^\frac 12\right \Vvert \leq \Vvert X-AXB\Vvert ,$ for all $X$ in $B(H)$ and for all unitarily invariant norms $\Vvert \cdot \Vvert .$
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