Berezin Radius Inequalities of Functional Hilbert Space Operators

Year 2023, , 107 - 116, 30.06.2023

Hamdullah Başaran , Mehmet Gürdal

https://doi.org/10.33401/fujma.1254301

Abstract

We investigate new upper bounds for the Berezin radius and Berezin norm of $2\times2$ operator matrices using the Cauchy-Buzano inequality, and we propose a required condition for the equality case in the triangle inequalities for the Berezin norms. We also show various Berezin radius inequalities for matrices with $2\times2$ operators.

Keywords

Berezin norm, Berezin radius, Functional Hilbert space

References

• [1] N. Aronzajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.
• [2] F. A. Berezin, Covariant and contravariant symbols for operators, Math. USSR-Izv., 6 (1972), 1117-1151.
• [3] H. Başaran, M. Gürdal, A. N. Güncan, Some operator inequalities associated with Kantorovich and Hölder-McCarthy inequalities and their applications, Turkish J. Math., 43(1) (2019), 523-532.
• [4] H. Başaran, M. Gürdal, Berezin number inequalities via inequality, Honam Math. J., 43(3) (2021), 523-537.
• [5] H. Başaran, V. Gürdal, Berezin radius and Cauchy-Schwarz inequality, Montes Taurus J. Pure Appl. Math., 5(3) (2023), 16-22.
• [6] H. Başaran, V. Gürdal, On Berezin radius inequalities via Cauchy-Schwarz type inequalities, Malaya J. Mat., 11(2) (2023), 127-141.
• [7] M. T. Garayev, M. W. Alomari, Inequalities for the Berezin number of operators and related questions, Complex Anal. Oper. Theory, 15(30) (2021), 1-30.
• [8] M. Gürdal, H. Başaran, A-Berezin number of operators, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 48(1) (2022), 77-87.
• [9] M. B. Huban, H. Bas¸aran, M. G¨urdal, New upper bounds related to the Berezin number inequalities, J. Inequal. Spec. Funct., 12(3) (2021), 1-12.
• [10] M. T. Karaev, Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory, 7 (2013), 983-1018.
• [11] W. Bani-Domi, F. Kittaneh, Norm and numerical radius inequalities for Hilbert space operators, Linear Multilinear Algebra, 69(5) (2021), 934-945.
• [12] S. S. Dragomir, Power inequalities for the numerical radius of a product of two operators in Hilbert spaces, Sarajevo J. Math., 5 (2009), 269–278.
• [13] F. Kittaneh, Norm inequalities for sums of positive operators, J. Operator Theory, 48 (2002), 95–103.
• [14] F. Kittaneh, H. R. Moradi, Cauchy-Schwarz type inequalities and applications to numerical radius inequalities, Math. Inequal. Appl., 23(3) (2020), 1117-1125.
• [15] M. T. Karaev, Berezin symbol, and invertibility of operators on the functional Hilbert spaces, J. Funct. Anal., 238 (2006), 181-192.
• [16] M. B. Huban, H. Başaran, M. Gürdal, Some new inequalities via Berezin numbers, Turk. J. Math. Comput. Sci., 14(1) (2022), 129-137.
• [17] M. B. Huban, H. Başaran, M. Gürdal, Berezin number inequalities via convex functions, Filomat, 36(7) (2022), 2333-2344.
• [18] H. Başaran, M. B. Huban, M. Gürdal, Inequalities related to Berezin norm and Berezin number of operators, Bull. Math. Anal. Appl., 14(2) (2022), 1-11.
• [19] M. Bakherad, Some Berezin number inequalities for operator matrices, Czechoslovak Math. J., 68(4) (2018), 997-1009.
• [20] M. Bakherad, M. T. Garayev, Berezin number inequalities for operators, Concr. Oper., 6(1) (2019), 33-43.
• [21] M. Bakherad, M. Hajmohamadi, R. Lashkaripour, S. Sahoo, Some extensions of Berezin number inequalities on operators, Rocky Mountain J. Math., 51(6) (2021), 1941-1951.
• [22] A. Abu-Omar, F. Kittaneh, Numerical radius inequalities for nn operator matrices, Linear Algebra Appl., 468 (2015), 18-26.
• [23] M. Bakherad, K. Shebrawi, Upper bounds for numerical radius inequalities involving off-diagonal operator matrices, Ann. Funct. Anal., 9(3) (2018), 297-309.
• [24] F. Kittaneh, Notes on some inequalities for Hilbert space operators, Publ. Res. Inst. Math. Sci., 24(2) (1988), 283-293.
• [25] J. Aujla, F. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl., 369 (2003), 217-233.
• [26] S. S. Sahoo, N. Das, D. Mishra, Berezin number and numerical radius inequalities for operators on Hilbert spaces, Adv. Oper. Theory, 5 (2020), 714-727.
• [27] M. L. Buzano, Generalizzatione della disuguaglianza di Cauchy-Schwarz, Rend. Sem. Mat. Univ. e Politec. Torino, 31(1971/73) (1974) 405-409.
• [28] V. Gürdal, H. Başaran, M. B. Huban, Further Berezin radius inequalities, Palest. J. Math., 12(1) (2023), 757–767.