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

Author(s): Danko R. Jocic
Journal: Proc. Amer. Math. Soc. 126 (1998), 2705-2711.
MSC (1991): Primary 47A30; Secondary 47B05, 47B10, 47B15
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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: 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
Copyright of article: Copyright 1998, American Mathematical Society

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