Advanced refinements of Berezin number inequalities

Abstract

For a bounded linear operator $A$ on a functional Hilbert space $\mathcal{H}\left( \Omega\right) $, with normalized reproducing kernel $\widehat {k}_{\eta}:=\frac{k_{\eta}}{\left\Vert k_{\eta}\right\Vert _{\mathcal{H}}},$ the Berezin symbol and Berezin number are defined respectively by $\widetilde{A}\left( \eta\right) :=\left\langle A\widehat{k}_{\eta},\widehat{k}_{\eta}\right\rangle _{\mathcal{H}}$ and $\mathrm{ber}(A):=\sup_{\eta\in\Omega}\left\vert \widetilde{A}{(\eta)}\right\vert .$ A simple comparison of these properties produces the inequality $\mathrm{ber}% \left( A\right) \leq\frac{1}{2}\left( \left\Vert A\right\Vert_{\mathrm{ber}}+\left\Vert A^{2}\right\Vert _{\mathrm{ber}}^{1/2}\right) $ (see [17]). In this paper, we prove further inequalities relating them, and also establish some inequalities for the Berezin number of operators on functional Hilbert spaces

Keywords

• Berezin symbol
• Berezin number
• functional Hilbert space
• positive operator

References

1. Alomari, M. W., Refinements of some numerical radius inequalities for Hilbert space operators, Linear Multilinear Algebra, 69(7) (2021), 1208-1223. https://doi.org/10.1080/03081087.2019.1624682
2. Alomari, M. W., Improvements of some numerical radius inequalities, Azerb. J. Math., 12(1) (2022), 124-137.
3. Aronzajn, N., Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404. https://doi.org/10.1090/S0002-9947-1950-0051437-7
4. Bakherad, M., Some Berezin number inequalities for operator matrices, Czechoslovak Math. J., 68(143:4) (2018), 997-1009. https://doi.org/10.21136/CMJ.2018.0048-17
5. Bakherad, M., Garayev, M. T., Berezin number inequalities for operators, Concr. Oper., 6(1) (2019), 33-43. http://doi.org/10.1515/conop-2019-0003
6. Bakherad, M., Hajmohamadi, M., Lashkaripour R., Sahoo, S., Some extensions of Berezin number inequalities on operators, Rocky Mountain J. Math., 51(6) (2021), 1941-1951. https://doi.org/10.1216/rmj.2021.51.1941
7. Başaran, H., Gürdal, M., Berezin number inequalities via inequality, Honam Math. J., 43(3) (2021)-523-537. https://doi.org/10.5831/HMJ.2021.43.3.523
8. Başaran, H., Huban, M. B., Gürdal, M., Inequalities related to Berezin norm and Berezin number of operators, Bull. Math. Anal. Appl., 14(2) (2022), 1-11. https://doi.org/10.54671/bmaa-2022-2-1

9. Berezin, F. A., Covariant and contravariant symbols for operators, Math. USSR-Izv., 6 (1972), 1117-1151. https://doi.org/10.1070/IM1972v006n05ABEH001913
10. Bhatia, R., Matrix Analysis, New York, Springer-Verlag, 1997.
11. Chalendar, I., Fricain, E., Gürdal, M., Karaev, M., Compactness and Berezin symbols, Acta Sci. Math. (Szeged), 78(1-2) (2012), 315-329. https://doi.org/10.1007/BF03651352
12. Garayev, M., Bouzeffour, F., Gürdal, M., Yangöz, C. M., Refinements of Kantorovich type, Schwarz and Berezin number inequalities, Extracta Math., 35 (2020), 1-20. https://doi.org/10.17398/2605-5686.35.1.1
13. Garayev, M. T., Guedri, H., Gürdal, M., Alsahli, G. M., On some problems for operators on the reproducing kernel Hilbert space, Linear Multilinear Algebra, 69(11) (2021), 2059-2077. https://doi.org/10.1080/03081087.2019.1659220
14. Gürdal, M., Başaran, H., A-Berezin number of operators, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 48(1) (2022), 77-87. https://doi.org/10.30546/2409-4994.48.1.2022.77
15. Hajmohamadi, M., Lashkaripour, R., Bakherad, M., Improvements of Berezin number inequalities, Linear Multilinear Algebra, 68(6) (2020), 1218-1229. https://doi.org/10.1080/03081087.2018.1538310
16. Haydarbeygi, Z., Sababbeb, M., Moradi H. R., A convex treatment of numerical radius inequalities, Czechoslovak Math. J., 72 (2022), 601–614. https://doi.org/10.21136/CMJ.2022.0068-21
17. Huban, M. B., Başaran, H., Gürdal, M., New upper bounds related to the Berezin number inequalities, J. Inequal. Spec. Funct., 12(3) (2021), 1-12.
18. Huban, M. B., Başaran, H., Gürdal, M., Some new inequalities via Berezin numbers, Turk. J. Math. Comput. Sci., 14(1) (2022), 129-137. https://doi.org/10.47000/tjmcs.1014841
19. Karaev, M. T., Berezin symbol and invertibility of operators on the functional Hilbert spaces, J. Funct. Anal., 238 (2006), 181-192. doi:10.1016/j.jfa.2006.04.030
20. Karaev, M. T., Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory, 7 (2013), 983-1018. https://doi.org/10.1007/s11785-012-0232-z
21. Kittaneh, F., Notes on some inequalities for Hilbert space operators, Publ. Res. Ins. Math. Sci., 24 (1988), 283-293. https://doi.org/10.2977/prims/1195175202
22. Kittaneh, F., Norm inequalities for sums and differences of positive operators, Linear Algebra Appl., 383 (2004), 85-91. https://doi.org/10.1016/j.laa.2003.11.023
23. Najafabadi, F. P., Moradi, H. R., Advanced refinements of numerical radius inequalities, Int. J. Math. Model. Comput., 11(4) (2021), 1-10. https://doi.org/10.30495/IJM2C.2021.684828
24. Omidvar, M. E., Moradi, H. R., Better bounds on the numerical radii of Hilbert space operators, Linear Algebra Appl., 604 (2020) 265-277. https://doi.org/10.1016/j.laa.2020.06.021
25. Omidvar, M. E., Moradi, H. R., Shebrawi, K., Sharpening some classical numerical radius inequalities, Oper. Matrices., 12(2) (2018), 407-416. doi:10.7153/oam-2018-12-26
26. Tafazoli, S., Moradi, H. R., Furuichi, S., Harikrishnan, P., Further inequalities for the numerical radius of Hilbert space operators, J. Math. Inequal., 13(4) (2019), 955-967. doi:10.7153/jmi-2019-13-68
27. Tapdigoglu, R., New Berezin symbol inequalities for operators on the reproducing kernel Hilbert space, Oper. Matrices, 15(3) (2021), 1445-1460. https://doi.org/10.7153/oam-2021-15-64
28. Yamancı, U., Tunç, R., Gürdal, M., Berezin number, Gr¨uss-type inequalities and their applications, Bull. Malays. Math. Sci. Soc., 43(3) (2020), 2287-2296. https://doi.org/10.1007/s40840-019-00804-x