A-Davis-Wielandt-Berezin radius inequalities

Abstract

We consider operator $V$ on the reproducing kernel Hilbert space $\mathcal{H}=\mathcal{H}(\Omega)$ over some set $\Omega$ with the reproducing kernel $K_{\mathcal{H},\lambda}(z)=K(z,\lambda)$ and define A-Davis-Wielandt-Berezin radius $\eta_{A}(V)$ by the formula $\eta_{A}(V):=sup\{\sqrt{| \langle Vk_{\mathcal{H},\lambda},k_{\mathcal{H},\lambda} \rangle_{A}|^{2}+\|Vk_{\mathcal{H},\lambda}\|_{A}^{4}}:\lambda \in \Omega\}$ and $\tilde{V}$ is the Berezin symbol of $V$ where any positive operator $A$-induces a semi-inner product on $\mathcal{H}$ is defined by $\langle x,y \rangle_{A}=\langle Ax,y \rangle$ for $x,y \in \mathcal{H}.$ We study equality of the lower bounds for A-Davis-Wielandt-Berezin radius mentioned above. We establish some lower and upper bounds for the A-Davis-Wielandt-Berezin radius of reproducing kernel Hilbert space operators. In addition, we get an upper bound for the A-Davis-Wielandt-Berezin radius of sum of two bounded linear operators.

Keywords

• Berezin symbol
• A-Davis-Wielandt-Berezin radius
• A-Berezin number
• A-Berezin norm
• semi inner product
• reproducing kernel Hilbert spaces.

References

1. Arias, M. L., Corach, G., Gonzales, M. C., Partial isometric in semi-Hilbertian spaces, Linear Algebra Appl., 428(7) (2008), 1460-1475. http://doi:10.1016/j.laa.2007.09.031
2. Arias, M. L., Corach, G., Gonzales, M. C., Metric properties of projection in semi-Hilbertian spaces, Integral Equ. Oper. Theory, 62 (2008), 11-28. http://doi:10.1007/S00020-008-1613-6
3. Bakherad, M., Garayev, M. T., Berezin number inequalities for operators, Concr. Oper., 6(1) (2019), 33-43. http://doi:10.1515/conop-2019-0003
4. 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
5. Bhunia, P., Bhanja, A., Bag, S., Paul, K., Bounds for the Davis-Wielandt radius of bounded linear operators, Ann. Funct. Anal., 12(18) (2021), 1-23. http://DOI: 10.1007/s43034-020-00102-9
6. Bhanja, A., Bhunia, P., Paul, K., On generalized Davis-Wielandt radius inequalities of semi-Hilbertian space operators, arXiv:2006.05069v1 [math.FA]. https://doi.org/10.48550/arXiv.2006.05069
7. Buzano, M. L., Generalizzatione della diseguaglianza di Cauchy-Schwarz, Rendiconti del Semin. Mat. dell Univ. di Padova, 31 (1971/73), 405-409.
8. Berezin, F. A., Covariant and contravariant symbols for operators, Math. USSR-Izvestiya, 6 (1972), 1117-1151. http://dx.doi.org/10.1070/IM1972v006n05ABEH001913

9. Chien, M. T., Nakazato, H., Davis-Wielandt shell and q-numerical range, Linear Algebra Appl., 340 (2002), 15-31. https://doi.org/10.1016/S0024-3795(01)00395-0
10. Davis, C., The shell of a Hilbert-space operator, Acta Sci. Math., 29(1-2) (1968), 69-86.
11. Dragomir, S. S., Reverses of Schwarz inequality in inner product spaces and applications, Math. Nachrichten, 288 (2015), 730-742. https://doi.org/10.1002/mana.201300100
12. Feki, K., A note on the A-numerical radius of operators in semi-Hilbert spaces, Arch. Math., 115(27) (2020), 535-544. https:doi.org/10.1007/s00013-020-01482-z
13. Feki K., Spectral radius of Semi-Hilbertian space operators and its applications, Ann. Funct. Anal., 11 (2020), 926-946. https:doi.org/10.1007/s43034-020-00064-y
14. Garayev, M. T., Alomari, M.W., Inequalities for the Berezin number of operators and related questions, Complex Anal. Oper. Theory, 15(30) (2021), 1-30. https://doi.org/10.1007/s11785-021-01078-7
15. Gürdal, M., Başaran, H., A-Berezin number of operators, Proc. Inst. Math. Mech., 48(1) (2022), 77-87. https://doi.org/10.30546/2409-4994.48.1.2022.77
16. Gürdal, V., Güncan, A. N., Berezin number inequalities via operator convex functions, Electron. J. Math. Anal. Appl., 10(2) (2022), 83-94.
17. Gürdal, V., Güncan, A. N., Upper and lower bounds for the Davis-Wielandt-Berezin radius, Preprint, (2021).
18. 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
19. Huban, M. B., Upper and lower bounds of the A-Berezin number of operators, Turk. J. Math., 46 (2022), 189-206. https://doi.org/10.3906/mat-2108-90
20. Karaev, M. T., Berezin symbol and invertibility of operators on the functional Hilbert spaces, J. Funct. Anal., 238 (2006), 181-192. https://doi.org/10.1016/j.jfa.2006.04.030
21. 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
22. Li, C. K., Poon, Y. T., Sze, N. S., Davis-Wielandt shells of operators, Oper. Matrices, 2(3) (2008), 341-355. https://doi.org/10.7153/oam-02-20
23. Moslehian, M. S., Kian, M., Xu, Q., Positivity of 2 × 2 block matrices of operators, Banach J. Math. Anal., 13(3) (2019), 726-743. https://doi.org/10.1215/17358787-2019-0019
24. Sattari, M., Moslehian, M. S., Shebrawi, K., Extension of Euclidean operator radius inequalities, Math. Scand., 120 (2017), 129-144. https://doi.org/10.7146/math.scand.a-25509
25. Wielandt, H., On eigenvalues of sums of normal matrices, Pac. J. Math., 5(4) (1955), 633-638. https://doi.org/10.2140/PJM.1955.5.633
26. Tapdigoglu, R., G¨urdal, 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
27. Zamani A., Characterization of numerical radius parallelism in $C^{*}$-algebras, Positivity, 23(2) (2019), 397-411. https://doi.org/10.1007/s11117-018-0613-2
28. Zamani, A., Shebrawi, K., Some upper bounds for the Davis-Wielandt radius of Hilbert space operators, Mediterr. J. Math., 17(25) (2020). https://doi.org/10.1007/s00009-019-1458-z