Hölder-type inequalities involving unitarily invariant norms

Hölder-type inequalities involving unitarily invariant norms General Hölder-type inequalities involving unitarily invariant norms for sums and products of Hilbert space operators are given. Among other inequalities, it is shown that if A, B and X are operators on a complex Hilbert space, then $$\left\vert \left\vert \left\vert {} \left\vert A^{\ast }XB\right\vert^{r} \right\vert \right\vert \right\vert ^{2}\leq \left\vert \left\vert \left\vert \left( A^{\ast }\left\vert X^{\ast} \right\vert A\right) ^{\frac{ pr}{2}} \right\vert \right\vert \right\vert ^{\frac{1}{p}} \left\vert \left\vert \left\vert \left( B^{\ast }\left\vert X\right\vert B\right) ^{ \frac{qr}{2}} \right\vert \right\vert \right\vert ^{\frac{1}{q}}$$ for all positive real numbers r, p and q such that p −1 + q −1 = 1 and for every unitarily invariant norm. The results in this article generalize some known Hölder inequalities for operators. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Hölder-type inequalities involving unitarily invariant norms

, Volume 16 (2) – May 3, 2011
16 pages

/lp/springer_journal/h-lder-type-inequalities-involving-unitarily-invariant-norms-x4uIQVRS06
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Fourier Analysis; Operator Theory; Econometrics; Potential Theory
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0122-z
Publisher site
See Article on Publisher Site

Abstract

General Hölder-type inequalities involving unitarily invariant norms for sums and products of Hilbert space operators are given. Among other inequalities, it is shown that if A, B and X are operators on a complex Hilbert space, then $$\left\vert \left\vert \left\vert {} \left\vert A^{\ast }XB\right\vert^{r} \right\vert \right\vert \right\vert ^{2}\leq \left\vert \left\vert \left\vert \left( A^{\ast }\left\vert X^{\ast} \right\vert A\right) ^{\frac{ pr}{2}} \right\vert \right\vert \right\vert ^{\frac{1}{p}} \left\vert \left\vert \left\vert \left( B^{\ast }\left\vert X\right\vert B\right) ^{ \frac{qr}{2}} \right\vert \right\vert \right\vert ^{\frac{1}{q}}$$ for all positive real numbers r, p and q such that p −1 + q −1 = 1 and for every unitarily invariant norm. The results in this article generalize some known Hölder inequalities for operators.

Journal

PositivitySpringer Journals

Published: May 3, 2011

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