Berezin inequalities for sums of operators and classical inequalities concerning the Berezin radius
Abstract
Introduction/purpose: In this article, the author’s goal is to seek to obtain new inequalities of the Berezin type.
Methods: The methods used are standard for operator theory.
Results: Various inequalities of the type given by Huban et al. and Erkan have been obtained.
Conclusions: In addition to obtaining various inequalities of the form given by Huban et al. and Erkan in particular, the authors sharpened the inequalities related to the Berezin norm.
References
Bakherad, M. & Garayev, M.T. 2019. Berezin number inequalities for operators. Concrete Operators, 6(1), pp.33-43. Available at: https://doi.org/10.1515/conop-2019-0003.
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), pp.1-11 [online]. Available at: https://www.emis.de/journals/BMAA/repository/docs/BMAA14-2-1.pdf [Accessed: 25 June 2024].
Berezin, F.A. 1972. Covariant and contavariant symbols of operators. Mathematics of the USSR-Izvestiya, 6(5), pp.1117-1151. Available at: https://doi.org/10.1070/IM1972v006n05ABEH001913.
Chalendar, I., Fricain, E., Gürdal, M. & Karaev, M. 2012. Compactness and Berezin symbols. Acta Scientiarum Mathematicarum, 78(1-2), pp.315-329. Available at: https://doi.org/10.1007/BF03651352.
Erkan, G.G. & Gürdal, M. 2024. Operatörlerin Berezin Yarıçap Eşitsizliklerinin İlerletilmesi. In: 3rd International Conference on Engineering, Natural and Social Sciences ICENSOS, Konya, Turkey, pp.463-470, May 16-17 [online]. Available at: https://www.icensos.com/3rdicensos2024 (in Turkish) [Accessed: 25 June 2024].
Furuta, T. 1994. An extension of the Heinz-Kato theorem. Proceedings of the American Mathematical Society, 120(3), pp.785-787. Available at: https://doi.org/10.2307/2160471.
Garayev, M.T. & Alomari, M.W. 2021. Inequalities for the Berezin number of operators and related questions. Complex Analysis and Operator Theory, 15, art.number:30. Available at: https://doi.org/10.1007/s11785-021-01078-7.
Garayev, M., Bouzeffour, F., Gürdal, M. & Yangöz, C.M. 2020. Refinements of Kantorovich type, Schwarz and Berezin number inequalitie. Extracta Mathemati- cae, 35(1), pp.1-20. Available at: https://doi.org/10.17398/2605-5686.35.1.1.
Garayev, M.T., Guediri, H., Gürdal, M. & Alsahli, G.M. 2021. On some problems for operators on the reproducing kernel Hilbert space. Linear and Multilinear Algebra, 69(11), pp.2059-2077. Available at: https://doi.org/10.1080/03081087.2019.1659220.
Güntürk, B. & Gürdal, M. 2024. On some refining inequalities via Berezin symbols. Honam Mathematical Journal, 46(3), pp.473-484. Available at: https://doi.org/10.5831/HMJ.2024.46.3.473.
Gürdal, V. & Başaran, H. 2023. On Berezin radius inequalities via Cauchy- Schwarz type inequalities. Malaya Journal of Matematik (MJM), 11(02), pp.127-141. Available at: https://doi.org/10.26637/mjm1102/002.
Gürdal, V., Başaran, H. & Huban, M.B. 2023. Further Berezin radius inequalities. Palestine Journal of Mathematics, 12(1), pp.757-767 [online]. Available at: https://pjm.ppu.edu/paper/1314-further-berezin-radius-inequalities [Accessed: 25 June 2024].
Gürdal, M. & Stojiljkovic, V. 2024a. Berezin radius inequalities for finite sums of functional Hilbert space operators. Gulf Journal of Mathematics, 17(1), pp.101-109. Available at: https://doi.org/10.56947/gjom.v17i1.1885.
Gürdal, M. & Stojiljkovic, V. 2024b. Some inequality and Berezin number type inequalities. Journal of Nonlinear Sciences and Applications, 17(4), pp.180-190. Available at: https://doi.org/10.22436/jnsa.017.04.02.
Gürdal, M. & Tapdigoglu, R. 2023. New Berezin radius upper bounds. Proceedings of the Institute of Mathematics and Mechanics, 49(2), pp.210-218. Available at: https://doi.org/10.30546/2409-4994.2023.49.2.210.
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), pp.1-12 [online]. Available at: http://www.ilirias.com/jiasf/repository/docs/JIASF12-3-1.pdf [Accessed: 25 June 2024].
Huban, M.B., Başaran, H. & Gürdal, M. 2022a. Some new inequalities via Berezin numbers. Turkish Journal of Mathematics and Computer Science, 14(1), pp.129-137. Available at: https://doi.org/10.47000/tjmcs.1014841.
Huban, M.B., Başaran, H. & Gürdal, M. 2022b. Berezin number inequalities via convex functions. Filomat, 36(7), pp.2333-2344. Available at: https://doi.org/10.2298/FIL2207333H.
Karaev, M.T. 2006. Berezin symbol and invertibility of operators on the functional Hilbert spaces. Journal of Functional Analysis, 238(1), pp.181-192. Available at: https://doi.org/10.1016/j.jfa.2006.04.030.
Kato, T. 1952. Notes on some inequalities for linear operators. Mathematische Annalen, 125, pp.208-212. Available at: https://doi.org/10.1007/BF01343117.
McCarthy, C.A. 1967. Cp. Israel Journal of Mathematics, 5, pp.249-271. Available at: https://doi.org/10.1007/BF02771613.
Singh Aujla, J. & Silva, F.C. 2003. Weak majorization inequalities and convex functions. Linear Algebra and its Applications, 369, pp.217-233. Available at: https://doi.org/10.1016/S0024-3795(02)00720-6.
Stojiljković, V. & Dragomir, S.S. 2024. Refinement of the Cauchy-Schwartz inequality with refinements and generalizations of the numerical radius type inequalities for operators. Annals of Mathematics and Computer Science, 21, pp.33-43. Available at: https://doi.org/10.56947/amcs.v21.246.
Stojiljković, V. & Gürdal, M. 2024a. Generalization of the Buzano’s inequality and numerical radius inequalities. Journal of Applied and Pure Mathematics, 6(3- 4), pp.191-200. Available at https://doi.org/10.23091/japm.2024.191.
Stojiljković, V. & Gürdal, M. 2024b. Berezin radius type inequalities for functional Hilbert space operators. Electronic Journal of Mathematics, 7, pp.35-44. Available at: https://doi.org/10.47443/ejm.2024.017.
Stojiljković, V. & Gürdal, M. 2024c. Numerical radius inequalities for two and one operator. Preprint.
Tapdigoglu, R., Gürdal, M., Altwaijry, N. & Sarı, N. 2021. Davis-Wielandt-Berezin radius inequalities via Dragomir inequalities. Operator and Matrices, 15(4), pp.1445-1460. Available at: https://doi.org/10.7153/oam-2021-15-90.
Yamancı, U., Tunç, R. & Gürdal, M. 2020. Berezin Number, Grüss-type inequalities and their applications. Bulletin of the Malaysian Mathematical Sciences Society, 43, pp.2287-2296. Available at: https://doi.org/10.1007/s40840-019-00804-x.
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