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Positivity

, Volume 16, Issue 2, pp 255–270 | Cite as

Hölder-type inequalities involving unitarily invariant norms

  • Hussien Albadawi
Article

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.

Keywords

Unitarily invariant norm Norm inequality n-tuple of operators Hölder’s inequality Cauchy-Schwarz inequality Positive operator 

Mathematics Subject Classification (2000)

47A30 47A63 47B10 47B15 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.King Faisal UniversityAhsaaSaudi Arabia

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