Advanced refinements of Berezin number inequalities

Year 2023, Volume: 72 Issue: 2, 386 - 396, 23.06.2023

Mehmet Gürdal , Hamdullah Başaran

https://doi.org/10.31801/cfsuasmas.1160606

Cited By: 1

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

• 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
• Alomari, M. W., Improvements of some numerical radius inequalities, Azerb. J. Math., 12(1) (2022), 124-137.
• Aronzajn, N., Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404. https://doi.org/10.1090/S0002-9947-1950-0051437-7
• 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
• 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
• 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
• 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
• 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
• Berezin, F. A., Covariant and contravariant symbols for operators, Math. USSR-Izv., 6 (1972), 1117-1151. https://doi.org/10.1070/IM1972v006n05ABEH001913
• Bhatia, R., Matrix Analysis, New York, Springer-Verlag, 1997.
• 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
• 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
• 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
• 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
• 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
• 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
• 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.
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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