AN ARITHMETIC-GEOMETRIC MEAN INEQUALITY RELATED TO NUMERICAL RADIUS OF MATRICES

Year 2017, Volume: 5 Issue: 1, 85 - 91, 01.04.2017

Alemeh Sheıkhhosseını

Abstract

For positive matrices $A, B \in \mathbb{M}_{n}$ and for all $X \in \mathbb{M}_{n}$, we show that $ \omega(AXA)\leq \frac{1}{2} \omega(A^{2}X+XA^{2}),$ and the inequality $ \omega(AXB) \leq \frac{1}{2}\omega(A^{2}X+XB^{2})$ does not hold in general, where $ \omega(.) $ is the numerical radius.

Keywords

Inequalities, Numerical radius, Unitarily invariant norms.

References

• [1] T. Ando, Matrix Young inequalities, Oper. Theory Adv. Appl. Vol: 75 (1995), 33-38.
• [2] T. Ando and K. Okubo, Induced norms of the Schur multiplication operator, Linear Algebra Appl. Vol:147 (1991), 181-199.
• [3] R. Bhatia, Positive Denite Matrices , Princeton University Press, 2007.
• [4] R. Bhatia and F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl. Vol:11 (1990), 272-277.
• [5] M. Erfanian Omidvar, M. Sal Moslehian and A. Niknam, Some numerical radius inequalities for Hilbert space operators, involve. Vol:2 (2009), 469-476.
• [6] K.E. Gustafson and D.K.M. Rao, Numerical Range, Springer-Verlag, New York, 1997.
• [7] C.R.Johnson, I. Spitkovsky and S. Gottlieb, Inequalities involving the numerical radius, Linear and Multilinear Algebra. Vol:37 (1994), 13-24.
• [8] A. Salemi and A. Sheikhhosseini, Matrix Young numerical radius inequalities, J. Math. Inequal. Vol:16, No.3 (2013), 783 -791.
• [9] A. Salemi and A. Sheikhhosseini, On reversing of the modied Young inequality, Ann. Funct. Anal. Vol:5, No.1 (2014), 69-75.
• [10] X. Zhan, Matrix Inequalities(Lecture notes in mathematics), Springer, New York, 2002.