A note on "More Operator Versions of the Schwarz Inequality"

A note on "More Operator Versions of the Schwarz Inequality" It is shown that for any (n + 1)-positive (possibly non-linear) map Φ and any bounded linear operators A i ,i = 1,¨,n we have [Φ(A i * A j )] i,j = 1 *≥[Φ(A i )*Φ(A j )] i,j = 1 *, and that the statement is false if "(n + 1)-positive" is replaced by "n-positive". This resolves an issue raised by Bhatia and Davis in relation to a Schwartz inequality which can be regarded as a non-commutative variance-covariance inequality [2] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

A note on "More Operator Versions of the Schwarz Inequality"

Positivity , Volume 8 (1) – Oct 21, 2004

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Publisher
Springer Journals
Copyright
Copyright © 2004 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/B:POST.0000023200.14261.fc
Publisher site
See Article on Publisher Site

Abstract

It is shown that for any (n + 1)-positive (possibly non-linear) map Φ and any bounded linear operators A i ,i = 1,¨,n we have [Φ(A i * A j )] i,j = 1 *≥[Φ(A i )*Φ(A j )] i,j = 1 *, and that the statement is false if "(n + 1)-positive" is replaced by "n-positive". This resolves an issue raised by Bhatia and Davis in relation to a Schwartz inequality which can be regarded as a non-commutative variance-covariance inequality [2]

Journal

PositivitySpringer Journals

Published: Oct 21, 2004

References

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