Cauchy-Schwarz and means inequalities for elementary operators into norm ideals

Author:
Danko R. Jocić

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2705-2711

MSC (1991):
Primary 47A30; Secondary 47B05, 47B10, 47B15

DOI:
https://doi.org/10.1090/S0002-9939-98-04342-1

MathSciNet review:
1451812

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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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Additional Information

**Danko R. Jocić**

Affiliation:
University of Belgrade, Faculty of Mathematics, Studentski trg 16, P. O. Box 550, 11000 Belgrade, Yugoslavia

Email:
jocic@matf.bg.ac.yu

Keywords:
Unitarily invariant norms,
Ky Fan dominance property.

Received by editor(s):
March 12, 1996

Received by editor(s) in revised form:
February 4, 1997

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1998
American Mathematical Society