Some upper bounds for the $$\mathbb {A}$$ -numerical radius of $$2\times 2$$ block matrices
Tóm tắt
Let
$$\mathbb {A}=\begin{bmatrix} A &{} 0 \\ 0 &{} A \end{bmatrix}$$
be the
$$2\times 2$$
diagonal operator matrix determined by a positive bounded linear operator A on a Hilbert space. For semi-Hilbertian operators X and Y, we first show that
$$\begin{aligned} w^2_{\mathbb {A}}\left( \begin{bmatrix} 0 &{} X \\ Y &{} 0 \end{bmatrix}\right)&\le \frac{1}{4}\max \Big \{{\big \Vert XX^{\sharp _A} + Y^{\sharp _A}Y\big \Vert }_{A}, {\big \Vert X^{\sharp _A}X + YY^{\sharp _A}\big \Vert }_{A}\Big \}\\&\quad + \frac{1}{2}\max \big \{w_{A}(XY), w_{A}(YX)\big \}, \end{aligned}$$
where
$$w_{\mathbb {A}}(\cdot )$$
,
$${\Vert \cdot \Vert }_{A}$$
and
$$w_{A}(\cdot )$$
are the
$$\mathbb {A}$$
-numerical radius, A-operator seminorm and A-numerical radius, respectively. We then apply the above inequality to find some upper bounds for the
$$\mathbb {A}$$
-numerical radius of certain
$$2\times 2$$
operator matrices. In particular, we obtain some refinements of earlier A-numerical radius inequalities for semi-Hilbertian operators. An upper bound for the
$$\mathbb {A}$$
-numerical radius of
$$2\times 2$$
block matrices of semi-Hilbertian space operators is also given.
Tài liệu tham khảo
Arias, M.L., Corach, G., Gonzalez, M.C.: Metric properties of projections in semi-Hilbertian spaces. Integr. Equ. Oper. Theory 62(1), 11–28 (2008)
Arias, M.L., Corach, G., Gonzalez, M.C.: Partial isometries in semi-Hilbertian spaces. Linear Algebra Appl. 428(7), 1460–1475 (2008)
Baklouti, H., Feki, K., Sid Ahmed, O.A.M.: Joint numerical ranges of operators in semi-Hilbertian spaces. Linear Algebra Appl. 555, 266–284 (2018)
Baklouti, H., Feki, K., Sid Ahmed, O.A.M.: Joint normality of operators in semi-Hilbertian spaces. Linear Multilinear Algebra 68(4), 845–866 (2020)
Bhunia, P., Feki, K., Paul, K.: Numerical radius parallelism and orthogonality of semi-Hilbertian space operators and its applications. Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-020-00392-8
Bhunia, P., Paul, K., Nayak, R.K.: On inequalities for \(A\)-numerical radius of operator. Electron. J. Linear Algebra 36, 143–157 (2020)
Buzano, M.L.: Generalizzazione della diseguaglianza di Cauchy–Schwarz, (Italian), Rend, Sem. Mat. Univ. e Politech. Torino 31, 405–409 (1974)
Douglas, R.G.: On majorization, factorization, and range inclusion of operators on Hilbert spaces. Proc. Amer. Math. Soc. 17, 413–415 (1966)
Feki, K.: Spectral radius of semi-Hilbertian space operators and its applications. Ann. Funct. Anal. 11, 929–946 (2020)
Feki, K., Sid Ahmed, O.A.M.: Davis-Wielandt shells of semi-Hilbertian space operators and its applications. Banach J. Math. Anal. 14, 1281–1304 (2020)
Hirzallah, O., Kittaneh, F., Shebrawi, K.: Numerical radius inequalities for certain \(2\times 2\) operator matrices. Integral Equ. Oper. Theory 71(1), 129–147 (2011)
Hirzallah, O., Kittaneh, F., Shebrawi, K.: Numerical radius inequalities for certain \(2\times 2\) operator matrices. Studia Math. 210(2), 99–114 (2012)
Majdak, W., Secelean, N.A., Suciu, L.: Ergodic properties of operators in some semi-Hilbertian spaces. Linear Multilinear Algebra 61(2), 139–159 (2013)
M.S. Moslehian and M. Sattari, Inequalities for operator space numerical radius of \(2\times 2\) block matrices, J. Math. Phys. 57 (2016), no. 1, 015201, 15pp
Moslehian, M.S., Xu, Q., Zamani, A.: Seminorm and numerical radius inequalities of operators in semi-Hilbertian spaces. Linear Algebra Appl. 591, 299–321 (2020)
Rout, N.C., Sahoo, S., Mishra, D.: Some \(A\)-numerical radius inequalities for semi-Hilbertian space operators. Linear Multilinear Algebra (2020). https://doi.org/10.1080/03081087.2020.1774487
Sahoo, S., Das, N., Mishra, D.: Numerical radius inequalities for operator matrices. Adv. Oper. Theory 4(1), 197–214 (2019)
Shebrawi, K.: Numerical radius inequalities for certain \(2\times 2\) operator matrices II. Linear Algebra Appl. 523, 1–12 (2017)
Suciu, L.: Maximum subspaces related to \(A\)-contractions and quasinormal operators. J. Korean Math. Soc. 45(1), 205–219 (2008)
Zamani, A.: \(A\)-numerical radius inequalities for semi-Hilbertian space operators. Linear Algebra Appl. 578, 159–183 (2019)