Banach Journal of Mathematical Analysis

Matrix inequalities related to Hölder inequality

Hussien Albadawi

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Matrix inequalities of Hölder type are obtained. Among other inequalities, it is shown that if $p,q \in (2,\infty) $ and $r>1$ with $1/p+1/q=1-1/r$, then for any $A_{i},B_{i}\in M_{n}\left(\mathbb{C} \right) $ and $\alpha _{i}\in \left[ 0,1\right] $ $\left( i=1,2,\cdots ,m\right) $ with $\sum\limits_{i=1}^{m}\alpha _{i}=1$, we have% \begin{equation*} \left\vert \sum\limits_{i=1}^{m}\alpha _{i}^{1/r}B_{i}A_{i}\right\vert \leq \left( \sum\limits_{i=1}^{m}\left\vert A_{i}\right\vert ^{p}\right) ^{1/p} \end{equation*}% whenever $\sum\limits_{i=1}^{m}\left\vert B_{i}^{\ast }\right\vert ^{q}\leq I $. Related unitarily invariant norm inequalities are also presented.

Article information

Banach J. Math. Anal., Volume 7, Number 2 (2013), 162 -171 .

First available in Project Euclid: 20 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A45
Secondary: 47A30 15A60 15B48

H\"{o}lder's inequality unitarily invariant norm norm inequality $m$-tuple of matrices


Albadawi , Hussien. Matrix inequalities related to Hölder inequality. Banach J. Math. Anal. 7 (2013), no. 2, 162 --171. doi:10.15352/bjma/1363784229.

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