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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Danko R. Jocic
Journal: Proc. Amer. Math. Soc. 126 (1998), 2705-2711
MSC (1991): Primary 47A30; Secondary 47B05, 47B10, 47B15
MathSciNet review: 1451812
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Abstract: The Cauchy-Schwarz norm inequality for normal elementary operators

\begin{displaymath}\left|\!\left|\!\left|\sum _{n=1}^\infty A_nXB_n \right|\!\right|\!\right|\leq \left|\!\left|\!\left| (\sum _{n=1}^\infty A_n^*A_n)^{1/2}X (\sum _{n=1}^\infty B_n^*B_n)^{1/2} \right|\!\right|\!\right|,\end{displaymath}

implies a means inequality for generalized normal derivations

\begin{displaymath}\left|\!\left|\!\left| \frac{AX+XB}2\right|\!\right|\!\right|\leq \left|\!\left|\!\left|X \right|\!\right|\!\right|^{1-\frac 1r} \left|\!\left|\!\left| \frac{|A|^rX+X|B|^r}2 \right| \!\right|\!\right|^\frac 1r,\end{displaymath}

for all $r\ge 2$, as well as an inequality for normal contractions $A$ and $B$

\begin{displaymath}\left|\!\left|\!\left| (I-A^*A) ^\frac 12X(I-B^*B)^\frac 12\right|\!\right|\!\right| \leq \left|\!\left|\!\left|X-AXB\right|\!\right|\!\right|, \end{displaymath}

for all $X$ in $B(H)$ and for all unitarily invariant norms $\left|\!\left|\!\left|\cdot \right|\!\right|\!\right|.$


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Additional Information

Danko R. Jocic
Affiliation: University of Belgrade, Faculty of Mathematics, Studentski trg 16, P. O. Box 550, 11000 Belgrade, Yugoslavia
Email: jocic@matf.bg.ac.yu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04342-1
PII: S 0002-9939(98)04342-1
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