Banach Journal of Mathematical Analysis

Matrix inequalities related to Hölder inequality

Hussien Albadawi

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 20 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1363784229

Digital Object Identifier
doi:10.15352/bjma/1363784229

Mathematical Reviews number (MathSciNet)
MR3039945

Zentralblatt MATH identifier
1270.15012

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

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

Citation

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


Export citation

References

  • H. Albadawi, Hölder-type inequalities involving unitarily invariant norms, Positivity 16 (2012), 255–270.
  • L. Arambašić, D. Bakić and M.S. Moslehian, A treatment of the Cauchy-Schwarz inequalityin $C^*$-modules, J. Math. Anal. Appl. 381 (2011), 546–556.
  • T. Ando and F. Hiai, Hölder type inequalities for matrices, Math. Inequal. Appl. 1 (1998), 1–30.
  • R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997.
  • R. Bhatia, Positive Definite Matrices, Princeton Press, 2007.
  • R. Bhatia and C. Davis, A Cauchy-Schwars inequality for operators with applications, Linear Algebra Appl. 223/224 (1995), 119–129.
  • S.S. Dragomir, M.S. Moslehian and J. Sandor, $Q$-norm inequalities for sequences of Hilbert space operators, J. Math. Inequal. 3 (2009), 1–14.
  • J.I. Fujii, Operator-valued inner product and operator inequalities, Banach J. Math. Anal. 2 (2008), 59–67.
  • J.I. Fujii, M. Fujii, M.S. Moslehian and Y. Seo, Cauchy-Schwarz inequality in semi-inner product $C^*$-modules via polar decomposition, J. Math. Anal. Appl. 394 (2012), 835–840.
  • F. Hiai and X. Zhan, Inequalities involving unitarily invariant norms and operator monotone functions, Linear Algebra Appl. 341 (2002), 151–169.
  • O. Hirzallah and F. Kittaneh, Inequalities for sums and direct sums of Hilbert space operators, Linear Algebra Appl. 424 (2007), 71–82.
  • R.A. Horn and R. Mathias, Cauchy-Schwarz inequality associated with positive semi definite matrices, Linear Algebra Appl. 142 (1990), 63–82.
  • R.A. Horn and X. Zhan, Inequalities for C-S semi norms and Lieb functions, Linear Algebra Appl. 291 (1999), 103–113.
  • D. Iliševic and S. Varošanec, On the Cauchy-Schwarz inequality and its reverse in semi-inner product $C^*$-modules, Banach J. Math. Anal. 1 (2007), 78–84.
  • T. Kato, Spectral order and a matrix limit theorem, Linear Multilinear Algebra 8 (1979), 15–19.
  • K. Shebrawi and H. Albadawi, Operator norm inequalities of Minkowski type, J. Inequal. Pure Appl. Math. 9 (2008), no. 1, Art. 26, 10 pp.
  • K. Shebrawi and H. Albadawi, Norm inequalities for the absolute value of Hilbert space operators, Linear and Multilinear Algebra 58 (2010), no. 4, 453–463.
  • K. Shebrawi and H. Albadawi, Trace inequalities for matrices, Bull. Austral. Math. Soc. 87 (2013), no. 1, 139–148.
  • X. Zhan, Matrix Inequalities, Springer-Verlag, Berlin, 2002.