Some Applications of Berezin Radius Inequalities

Year 2025, Volume: 12 Issue: 2, 562 - 582, 30.06.2025

Hamdullah Başaran

https://doi.org/10.54287/gujsa.1655950

Abstract

We show a number of inequalities in the reproducing kernel Hilbert space (RKHS) in this study. Stronger boundaries between the Berezin radius and the numerical radius on the Berezin number of the bounded linear operator described in the RKHS than those found in the literature are obtained by using a few auxiliary theorems while analyzing the Berezin radius.

Keywords

Reproducing Kernel Hilbert Space , Operator Norm , Berezin Symbol , Polar Decomposition , Contraction Operator

References

• Abu-Omar, A., & Kittaneh F. (2015). Upper and lower bounds for the numerical radius with an application to involution operators. The Rocky Mountain Journal of Mathematics, 45(4), 1055-065. https://doi.org/10.1216/RMJ-2015-45-4-1055
• Aronzajn, N. (1950). Theory of reproducing kernels. Transactions of the American Mathematical Society, 68, 337-404.
• Başaran H. (2024). Berezin radius inequalities related to the contraction operator. Montes Taurus Journal of Pure and Applied Mathematics, 6 (1), 102-109. Article ID:MTJPAM-D-23-00044.
• Başaran, H., Gürdal, M., & Güncan, A. N. (2019). Some operator inequalities associated with Kantorovich and Hö lder-McCarthy inequalities and their applications. Turkish Journal of Mathematics, 43(1) (2019), 523-532. http://doi.org/10.3906/mat-1811-10
• Başaran, H., & Gürdal, V. (2023). Berezin radius and Cauchy-Schwarz inequality. Montes Taurus Journal of Pure and Applied Mathematics, 5(3), 16-22. Article ID: MTJPAM-D-21-00068
• Başaran H., Huban M. B., Gürdal M. (2022). Inequalities related to Berezin norm and Berezin number of operators. Bulletin of Mathematical Analysis and Applications, 14(2), 1-11. https://doi.org/10.54671/bmaa-2022-2-1
• Berezin, F.A. (1972). Covariant and contravariant symbols for operators. Mathematics of the USSR-Izvestiya, 6(5), 1117-1151. http://dx.doi.org/10.1070/IM1972v006n05ABEH001913
• Bhunia, P., Dragomir, S. S., Moslehian, M. S., & Paul, K. (2022). Lectures on Numerical Radius Inequalities. Springer Cham.
• Bhatia, R. (2007). Positive define matrices. Princeton, NJ: Princeton University Press.
• Bhunia, P., Jana, S., & Paul, K. (2023a). Numerical radius inequalities and estimation of zeros of polynomials. Georgian Mathematical Journal, 30(5), 671-682. https://doi.org/10.1515/gmj-2023-2037
• Bhunia, P., Gürdal, M., Paul, K., Sen, A., & Tapdigoglu, R. (2023b). On a new norm on the space of reproducing kernel Hilbert space operators and Berezin radius inequalities. Numerical Functional Analysis and Optimization, 44(9):970-986. https://doi.org/10.1080/01630563.2023.2221857
• Buzano, M. L. (1974). Generalizzatione della disuguaglianza di Cauchy-Schwarz. Rendiconti del Seminario Matematico. Universit_a e Politecnico Torino, 31 (1971/73), 405-409.
• Dragomir, S. S. (2016). Buzano's inequality holds for any projection. Bulletin of the Australian Mathematical Society, 3, 504-510. https://doi.org/10.1017/S0004972715001525
• Engliš, M. (1995). Toeplitz operators and the Berezin transform on H^2. Linear Algebra and Applications, 223-224, 171-204. https://doi.org/10.1016/0024-3795(94)00056-J
• Furuta, T. (2001). Invilation to linear operators, London: Taylor and Francis.
• Garayev, M. T., Guedri, H., Gürdal M., & Alsahli, G. M. (2021). On some problems for operators on the reproducing kernel Hilbert space. Linear and Multilinear Algebra, 69(11), 2059-2077. https://doi.org/10.1080/03081087.2019.1659220
• Gürdal, M., Alomari, M.W., Başaran, H. (2025). Berezin radius inequalities via Orlicz function. Palestine Journal of Mathematics, 14(1), 928-945.
• Gürdal, M., Erkan, G. G., & Garayev, M. (2024). Berezin norm and Berezin radius inequalities of product and sums with Selberg inequality. Proceedings of the Institute of Mathematics and Mechanics, 50(2), 258-273. https://doi.org/10.30546/2409-4994.2024.50.2.258
• Gürdal, V., & Başaran H. (2023). On Berezin radius inequalities via Cauchy-Schwarz type inequalities. Malaya Journal of Matematik, 11(02) (2023), 127-141. https://doi.org/10.26637/mjm1102/002
• Huban, M.B., Başaran, H. & Gürdal, M. (2021). New upper bounds related to the Berezin number inequalities. Journal of Inequalities and Special Functions, 12(3), 1-12.
• Huban, M.B., Başaran, H. & Gürdal, M. (2022a). Some new inequalities via Berezin numbers. Turkish Journal of Mathematics and Computer Science, 14(1), 129-137. https://doi.org/10.47000/tjmcs.1014841
• Huban, M.B., Başaran, H. & Gürdal, M. (2022b). Berezin number inequalities via convex functions. Filomat, 36(7), 2333-2344. https://doi.org/10.2298/FIL2207333H
• Jorgensen, P. (2006). Analysis and probability: wavelets, signals, fractals, Spinger.
• Karaev, M.T. (2006). Berezin symbol and invertibility of operators on the functional Hilbert spaces. Journal of Functional Analysis, 238, 181-192. https://doi.org/10.1016/j.jfa.2006.04.030
• Karaev, M.T. (2013). Reproducing kernels and Berezin symbols techniques in various questions of operator theory. Complex Analysis and Operator Theory, 7, 983-1018. https://doi.org/10.1007/s11785-012-0232-z
• Karaev, M.T., Gürdal, M., & Huban, M.B. (2016). Reproducing kernels, Englis algebras and some applications. Studia Mathematica, 232(2), 113-141. https://doi.org/10.4064/sm8385-4-2016
• Karaev, M., Gürdal, M., & Saltan, S. (2011). Some applications of Banach algebra techniques. Mathematische Nachrichten, 284(13):1678-1689. https://doi.org/10.1002/mana.200910129
• Kato, T. (1952). Notes on some inequalities for linear operators. Mathematische Annalen, 125, 208-212. https://doi.org/10.1007/BF01343117
• Kittaneh, F. (1988). Notes on some inequalities for Hilbert space operators. Publications of the Research Institute for Mathematical Sciences, 24, 283-293. https://doi.org/10.2977/PRIMS/1195175202
• Kittaneh, F. (1997). Norm inequalities for certain operators sums. Journal of Functional Analysis, 143, 337-348. https://doi.org/10.1006/jfan.1996.2957
• Kittaneh, F., Moslehian, M. S., & Yamakazi, T. (2015). Caretesian decomposition and numerical radius inequalities. Linear Algebra and Applications, 471, 46-53. https://doi.org/10.1016/j.laa.2014.12.016
• Pečarić, J., Furuta, T., Hot, M.H., & Seo, Y. (2005). Mond-Pečarić method in operator inequalities. Zagreb; Elements.
• Sababheh, M., Conde, C., & Moradi, H. R. (2023). On the Cauchy-Schwarz inequality. Operators and Matrisces, 17(2) (2023), 525-538. https://dx.doi.org/10.7153/oam-2023-17-34
• Sababheh, M., & Moradi, H. R. (2023). Numerical radius of kronecker product of matrices. Journal of Applied Analysis & Computation, 16(3), 2943-2954. https://www.jaac-online.com/article/doi/10.11948/20230064
• Sababheh, M., Moradi, H. R., & Sahoo, S. (2024). Inner product inequalities with application. Linear and Multilinear Algebra, (2024), 1-14. https://doi.org/10.1080/03081087.2024.2312444
• Stojiljkovic, V., & Gürdal, M. (2025). Estimates of the numerical radius utilizing various function properties. Turkish Journal of Mathematics, 49(1), 49-64. https://doi.org/10.55730/1300-0098.3573
• Yamakazi, T. (2007). On upper and lower bounds of the numerical radius and an equality condition. Studia Mathematica, 178, 83-89. https://doi.org/10.4064/sm178-1-5
• Yamancı, U., Tunç, R., & Gürdal, M. (2020). Berezin number, Grüss-type inequalities and their applications. Bulletin of the Malaysian Mathematical Sciences Society, 43(3), 2287-2296. https://doi.org/10.1007/s40840-019-00804-x
• Yang, C., & Xu, M. (2023). Some new numerical radius and Hilbert-Schmitd numerical radius inequalities for Hilbert space operators. Journal of Mathematical Inequalities, 17(1), 269-282. https://dx.doi.org/10.7153/jmi-2023-17-19
• Zhu, K., (1990). Operator Theory in Function Spaces, Marcel Dekker Inc., New York, USA Series: Pure and Applied Mathematicsa Series of Monographs and Textbooks, Volume 139.