Numerical radius inequalities for operator matrices. (English) Zbl 06946450

Summary: Several numerical radius inequalities for operator matrices are proved by generalizing earlier inequalities. In particular, the following inequalities are obtained: if \(n\) is even, \[ 2w(T) \leq \max\{\| A_1 \|, \| A_2 \|,\dots,\| A_n \|\}+\frac{1}{2}\sum_{k=0}^{n-1} \| |A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} \|, \] and if \(n\) is odd, \[ 2w(T) \leq \max\{\| A_1 \|,\| A_2 \|,\dots,\| A_n \|\}+w\bigg(\widetilde{A}_{(\frac{n+1}{2})t}\bigg)+ \frac{1}{2}\sum_{k=0}^{n-1} \| |A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} \|, \] for all \(t\in [0,1]\), \( A_i\)’s are bounded linear operators on the Hilbert space \(\mathcal{H}\), and \(T\) is off diagonal matrix with entries \(A_1, \cdots, A_n\).


47A12 Numerical range, numerical radius
47A63 Linear operator inequalities
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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[1] A. Abu-Omar and F. Kittaneh, A numerical radius inequality involving the generalized Aluthge transform, Studia Math. 216 (2013), 69–75. · Zbl 1279.47015
[2] M. Bakherad and K. Shebrawi, Some generalizations of the Aluthge transform of operators and their consequences, preprint.
[3] R. Bhatia, Matrix analysis, Graduate Texts in Mathematics, 169, Springer-Verlag, New York, 1997. · Zbl 0863.15001
[4] J. F. Carlson, D. N. Clark, C. Foias, and J. P. Williams, Projective Hilbert \(\bold A(\bold D)\)-modules, New York J. Math. 1 (1994/95), 26–38.
[5] S. Dragomir, A survey of some recent inequalities for the norm and numerical radius of operators in Hilbert spaces, Banach J. Math. Anal. 2 (2007), no. 1, 154–175. · Zbl 1136.47006
[6] P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933. · Zbl 0204.15001
[7] O. Hirzallah, F. Kittaneh, and K. Shebrawi, Numerical radius inequalities for certain \(2× 2\) operator matrices, Integr. Equ. Oper. Theory. 71 (2011), 129–147. · Zbl 1238.47004
[8] J. C. Hou and H. K. Du, Norm inequalities of positive operator matrices, Integr. Equ. Oper. Theory. 22 (1995), 281–294. · Zbl 0839.47004
[9] F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158 (2003), no. 1, 11–17. · Zbl 1113.15302
[10] A. D. Mohammed, A. Z. Khaldoun, A. Mohammed, and B. A. Feras, General numerical radius inequalities for matrices of operators, Open Math. 14 (2016), 109–117. · Zbl 1354.47006
[11] K. Okubo, On weakly unitarily invariant norm and the Aluthge transformation, Linear Algebra Appl. 371 (2003), 369–375. · Zbl 1029.15025
[12] S. Petrovic, Some remarks on the operator of Foias and Williams, Proc. Amer. Math. Soc. 124 (1996), no. 9, 2807–2811. · Zbl 0859.47020
[13] K. Shebrawi, Numerical Radius inequalities for certain \(2×2\) operator matrices II, Linear Algebra Appl. 523 (2017), 1–12. · Zbl 1453.47001
[14] K. Shebrawi and H. Albadawi, Numerical radius and operator norm inequalities, J. Inequal. Appl. 1 (2009), Article ID 492154. · Zbl 1179.47004
[15] T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 178 (2007), no. 1, 83–89. · Zbl 1114.47003
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