Some refinements of Berezin number inequalities via convex functions

Abstract

The Berezin transform $\widetilde{A}$ and the Berezin number of an operator $A$ on the reproducing kernel Hilbert space over some set $\Omega$ with normalized reproducing kernel $\widehat{k}_{\lambda}$ are defined, respectively, by $\widetilde{A}(\lambda)=\left\langle {A}\widehat{k}_{\lambda },\widehat{k}_{\lambda}\right\rangle ,\ \lambda\in\Omega$ and $\mathrm{ber}% (A):=\sup_{\lambda\in\Omega}\left\vert \widetilde{A}{(\lambda)}\right\vert .$ A straightforward comparison between these characteristics yields the inequalities $\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) $. In this paper, we study further inequalities relating them. Namely, we obtained some refinement of Berezin number inequalities involving convex functions. In particular, for $A\in\mathcal{B}\left( \mathcal{H}% \right) $ and $r\geq1$ we show that \[ \mathrm{ber}^{2r}\left( A\right) \leq\frac{1}{4}\left( \left\Vert A^{\ast }A+AA^{\ast}\right\Vert _{\mathrm{ber}}^{r}+\left\Vert A^{\ast}A-AA^{\ast }\right\Vert _{\mathrm{ber}}^{r}\right) +\frac{1}{2}\mathrm{ber}^{r}\left( A^{2}\right) . \]

Keywords

• Berezin number
• Berezin transform
• Cauchy-Schwarz inequality
• Young inequality
• reproducing kernel Hilbert space

References

1. Abu-Omar, A., Kittaneh, F., Upper and lower bounds for the numerical radius with an application to involution operators, Rocky Mountain J. Math., 45 (2015), 1055-1065. https://doi.org/10.1216/RMJ-2015-45-4-1055
2. Aronzajn, N., Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.
3. Bakherad, M., Some Berezin number inequalities for operator matrices, Czechoslovak Mathematical Journal, 68(4) (2018), 997-1009. http://doi.org/10.21136/CMJ.2018.0048-17
4. Bakherad, M., Garayev, M. T., Berezin number inequalities for operators, Concrete Operators, 6(1) (2019), 33-43. http://doi.org/10.1515/conop-2019-0003
5. Başaran, H., Gürdal, M., Güncan, A. N., Some operator inequalities associated with Kantorovich and Hölder-McCarthy inequalities and their applications, Turkish J. Math., 43(1) (2019), 523-532. http://doi.org/10.3906/mat-1811-10
6. Bellman, R., Almost orthogonal series, Bull. Amer. Math. Soc., 50 (1944), 517-519.
7. Berezin, F. A., Covariant and contravariant symbols for operators, Math. USSR-Izvestiya, 6 (1972), 1117-1151. http://dx.doi.org/10.1070/IM1972v006n05ABEH001913
8. Boas, R. P., A general moment problem, Amer. J. Math., 63 (1941), 361-370. https://doi.org/10.2307/2371530

9. Dragomir, S. S., Some refinements of Schwarz inequality, Suppozionul de Matematica si Aplicatii, Poly Technical Institute Timisoara, Romania, (1985), 13-16.
10. Dragomir, S. S., Some inequalities for the norm and the numerical radius of linear operators in Hilbert Spaces, Tamkang J. Math., 39 (2008), 1-7. https://doi.org/10.5556/j.tkjm.39.2008.40
11. Garayev, M. T., Berezin symbols, Hölder-McCarthy and Young inequalities and their applications, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 43(2) (2017), 287-295.
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., Gürdal, M., Okudan, A., Hardy-Hilbert’s inequality and a power inequality for Berezin numbers for operators, Math. Inequal. Appl., 19 (2016), 883-891. https://doi.org/10.7153/mia-19-64
14. 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.10.1080/03081087.2019.1659220
15. Gustafson, K. E., Rao, D. K. M., Numerical Range, Springer-Verlag, New York, 1997. https://doi.org/10.1007/978-1-4613-8498-4 1
16. Huban, M. B., Upper and lower bounds of the A-Berezin number of operators, Turkish J. Math., 46(1) (2022), 189-206. https://doi.org/10.3906/mat-2108-90
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., Berezin number inequalities via convex functions, Filomat, 36(7) (2022), 2333-2344. https://doi.org/10.2298/FIL2207333H
19. Karaev, M. T., Berezin symbol and invertibility of operators on the functional Hilbert spaces, J. Funct. Anal., 238 (2006), 181-192. https://doi.org/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.
21. Kittaneh, F., A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math., 158 (2003), 11-17. https://doi.org/10.4064/sm158-1-2
22. Kittaneh, F., Norm inequalities for sums and diferences of positive operators, Linear Algebra Appl., 383 (2004), 85-91. https://doi.org/10.1016/j.laa.2003.11.023
23. Kittaneh, F., Notes on some inequalities for Hilbert Space operators, Publ. Res. Inst. Math. Sci., 24 (1988), 283-293. https://doi.org/10.2977/prims/1195175202
24. Mond, B., Pecaric, J., On Jensen’s inequality for operator convex functions, Houston J. Math., 21 (1995), 739-753.
25. Moradi, H. R., Sababheh, M., More accurate numerical radius inequalities (II), Linear Multilinear Algebra, 69(5) (2021), 921-933. https://doi.org/10.1080/03081087.2019.1703886
26. Moradi, H. R. Omidvar, M. E., Dragomir, S. S., Khan, M. S., Sesquilinear version of numerical range and numerical radius, Acta Univ. Sapientiae Math., 9(2) (2017), 324-335. https://doi.org/10.1515/ausm-2017-0024
27. Omidvar, M. E., Moradi, H. R., Shebrawi, K., Sharpening some classical numerical radius inequalities, Oper. Matrices, 12(2) (2018), 407-416. https://doi.org/10.7153/oam-2018-12-26
28. Sababheh, M., Moradi, H. R., More accurate numerical radius inequalities (I), Linear Multilinear Algebra, 69(10) (2021). https://doi.org/10.1080/03081087.2019.1651815
29. Safshekan, R., Farokhinia, A., Some refinements of numerical radius inequalities via convex functions, Applied Mathematics E-Notes, 21 (2021), 542-549.
30. Sattari, M., Moslehian, M. S., Yamazaki, T., Some generalized numerical radius inequalities for Hilbert space operators, Linear Algebra Appl., 470 (2015), 216-227. https://doi.org/10.1016/j.laa.2014.08.003
31. Tafazoli, S., Moradi, H. R., Furuichi, S., Harikrishnan, P., Further inequalities for the numerical radius of Hilbert space operators, J. Math. Inequal., 13 (2019), 955-967. https://doi.org/10.48550/arXiv.1907.06003
32. 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
33. Tapdigoglu, T., Gürdal, M., Altwaijry, N., Sarı, N., Davis-Wielandt-Berezin radius inequalities via Dragomir inequalities, Oper. Matrices, 15(4) (2021), 1445-1460. https://doi.org/10.7153/oam-2021-15-90