Some bounds for the $\mathbb{A}$-numerical radius of certain $2 \times 2$ operator matrices

Abstract

For a given bounded positive (semidefinite) linear operator $A$ on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$, we consider the semi-Hilbertian space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle_A \big)$ where ${\langle x, y\rangle}_A := \langle Ax, y\rangle$ for every $x, y\in\mathcal{H}$. The $A$-numerical radius of an $A$-bounded operator $T$ on $\mathcal{H}$ is given by
\[\omega_A(T)=\sup\Big\{\big|{\langle Tx, x\rangle}_A\big|\,;\,\, x\in\mathcal{H},\, {\langle x, x\rangle}_A=1\Big\}.\]
Our aim in this paper is to derive several $\mathbb{A}$-numerical radius inequalities for $2\times 2$ operator matrices whose entries are $A$-bounded operators, where $\mathbb{A}=\text{diag}(A,A)$.

Keywords

• positive operator
• operator matrix
• semi-inner product
• $A$-numerical radius
• inequality

References

1. [1] M.L. Arias, G. Corach and M.C. Gonzalez, Partial isometries in semi-Hilbertian spaces, Linear Algebra Appl. 428 (7) 1460–1475, 2008.
2. [2] M.L. Arias, G. Corach and M.C. Gonzalez, Metric properties of projections in semi- Hilbertian spaces, Integral Equations Operator Theory, 62, 11–28, 2008.
3. [3] M.L. Arias, G. Corach and M.C. Gonzalez, Lifting properties in operator ranges, Acta Sci. Math. (Szeged) 75:3-4, 635–653, 2009.
4. [4] A. Abu-Omar and F. Kittaneh, Numerical radius inequalities for $n\times n$ operator matrices, Linear Algebra Appl. 468, 18–26, 2015.
5. [5] P. Bhunia, K. Feki and K. Paul, A-Numerical radius orthogonality and parallelism of semi-Hilbertian space operators and their applications, Bull. Iran. Math. Soc. 47, 435-457, 2021.
6. [6] H. Baklouti, K. Feki and O.A.M. Sid Ahmed, Joint numerical ranges of operators in semi-Hilbertian spaces, Linear Algebra Appl. 555, 266–284, 2018.
7. [7] H. Baklouti, K. Feki and O.A.M. Sid Ahmed, Joint normality of operators in semi- Hilbertian spaces, Linear Multilinear Algebra, 68 (4) 845–866, 2020.
8. [8] P. Bhunia, K. Paul and R.K. Nayak, On inequalities for A-numerical radius of oper- ators, Electron. J. Linear Algebra, 36, 143–157, 2020.

9. [9] P. Bhunia, K. Paul and R.K. Nayak, Sharp inequalities for the numerical radius of Hilbert space operators and operator matrices, Math. Inequal. Appl. 24 (1), 167–183, 2021.
10. [10] L. de Branges and J. Rovnyak, Square Summable Power Series, Holt, Rinehert and Winston, New York, 1966.
11. [11] R.G. Douglas, On majorization, factorization and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc. 17, 413–416, 1966.
12. [12] M. Faghih-Ahmadi and F. Gorjizadeh, A-numerical radius of A-normal operators in semi-Hilbertian spaces, Ital. J. Pure Appl. Math.- N. 36, 73–78, 2016.
13. [13] K. Feki, Spectral radius of semi-Hilbertian space operators and its applications, Ann. Funct. Anal. 11, 929–946, 2020.
14. [14] K. Feki, A note on the A-numerical radius of operators in semi-Hilbert spaces, Arch. Math. 115, 535–544, 2020.
15. [15] K. Feki, On tuples of commuting operators in positive semidefinite inner product spaces, Linear Algebra Appl. 603, 313–328, 2020.
16. [16] K. Feki, Generalized numerical radius inequalities of operators in Hilbert spaces, Adv. Oper. Theory 6 Article number:6, 2021.
17. [17] K. Feki, Some A-spectral radius inequalities for A-bounded Hilbert space operators, arXiv: 2002.02905 [math.FA].
18. [18] K. Feki and O.A.M. Sid Ahmed, Davis-Wielandt shells of semi-Hilbertian space op- erators and its applications, Banach J. Math. Anal. 14, 1281–1304, 2020.
19. [19] O. Hirzallah, F. Kittaneh and K. Shebrawi, Numerical radius inequalities for $2\times 2$ operator matrices, Studia Mathematica, 210, 99–115, 2012.
20. [20] A.Saddi, A-Normal operators in Semi-Hilbertian spaces, Aust. J. Math. Anal. Appl. 9 (1), 1–12, 2012.
21. [21] K. Shebrawi, Numerical radius inequalities for certain 2 × 2 operator matrices II, Linear Algebra Appl. 523, 1–12, 2017.
22. [22] T-Y Tam and P. Zhang, Spectral decomposition of selfadjoint matrices in positive semidefinite inner product spaces and its applications, Linear Multilinear Algebra, 67 (9), 1829–1838, 2019.
23. [23] A. Zamani, A-numerical radius inequalities for semi-Hilbertian space operators, Lin- ear Algebra Appl. 578, 159–183, 2019.