Specht Oranına Göre Berezin Sayı Eşitsizlikleri

Yıl 2022, , 1201 - 1214, 31.12.2022

Mehmet Gürdal , Hamdullah Başaran

https://doi.org/10.31202/ecjse.1131830

Öz

Berezin dönüşümü, düzgün fonksiyonları, analitik fonksiyonların Hilbert uzayları üzerindeki operatörlerle ilişkilendirir. Hilbert fonksiyonel uzay H(Ω) üzerinde bir A operatörünün Berezin sembolü ve Berezin sayısı
A ̃(μ)=〈A K_μ/K_μ ,K_μ/K_μ 〉,μ∈Ω ve ber(A)=sup┬(μ∈Ω)⁡|A ̃(μ)|
şeklinde tanımlanır. Bu A ̃ sınırlı fonksiyonu kullanılarak Hilbert fonksiyonel uzay operatörlerinin bazı yeni Berezin sayı eşitsizliklerini sunulmuştur. Specht oranı yardımıyla bazı eşitsizlikler genelleştirilmiş ve iyileştirilmiştir. Aynı zamanda bu iyileştirmeler kullanılarak Berezin yarıçap ve Berezin norm için çeşitli yeni eşitsizlikler gösterilmiştir.

Anahtar Kelimeler

Berezin sayısı, Hilbert fonksiyonel uzay, Specht oranı, pozitif operatör

Berezin number inequalities in terms of Specht's

Öz

Smooth functions are associated with operators on Hilbert spaces of analytic functions through the Berezin transform. The Berezin symbol and the Berezin number of an operator A on the Hilbert functional space H(Ω) over some set Ω with the reproducing kernel are defined, respectively, by
A ̃(μ)=〈A K_μ/K_μ ,K_μ/K_μ 〉,μ∈Ω and ber(A)=sup┬(μ∈Ω)⁡|A ̃(μ)|.
By using this bounded function A ̃, we present some new Berezin number inequalities of Hilbert functional space operators. Some inequalities with respect to Specht's ratio are improved and generalized. Using these modifications, we also establish various new inequalities for the Berezin radius and Berezin norm of operators.

Anahtar Kelimeler

Berezin number, Hilbert functional space, Specht, positive operator

Kaynakça

• Aronzajn, N., Theory of reproducing kernels, Trans. Amer. Math. Soc., 1950, 68, 337-404.
• Aujla, J., Silva, F., Weak majorization inequalities and convex functions, Linear Algebra Appl., 2003, 369, 217-233.
• Berezin, F.A., Covariant and contravariant symbols for operators, Math. USSR-Izvestiya, 1972, 6, 1117-1151.
• Bakherad, M., Garayev, M.T., Berezin number inequalities for operators, Concrete Operators 2019, 6(1), 33-43.
• Başaran, H., Gürdal, M., Berezin number inequalities via inequality, Honam Math. J., 2021, 43(3), 523-537.
• Başaran, H., Gürdal, V., Berezin radius and Cauchy-Schwarz inequality, Montes Taurus J. Pure Appl. Math., 2023, 5(3), 16-22.
• Başaran, H., Huban, M.B., Gürdal, M., Inequalities related to Berezin norm and Berezin number of operators, Bull. Math. Anal. Appl., 2022, 14(2), 1-11.
• Dragomir, S.S., On some inequalities for numerical radius of operators in Hilbert sapaces, Korean J. Math., 2017, 25(2), 247-259.
• Furuichi, S., Refined Young inequalities with Specht's ratio, J. Egyptian Math. Soc., 2012, 20(1), 46-49.
• Garayev, M., Bouzeffour, F., Gürdal, M., Yangöz, C.M., Refinements of Kantorovich type, Schwarz and Berezin number inequalities, Extracta Math., 2020, 35, 1-20.
• Garayev, M.T., Gürdal, M., Okudan, A., Hardy-Hilbert's inequality and a power inequality for Berezin numbers for operators, Math. Inequal. Appl., 2016, 19, 883-891.
• Garayev, M.T., Gürdal, M., Saltan, S., Hardy type inequaltiy for reproducing kernel Hilbert space operators and related problems, Positivity, 2017, 21, 1615-1623.
• Garayev, M.T., Guedri, H., Gürdal, M., Alsahli, G.M., On some problems for operators on the reproducing kernel Hilbert space, Linear Multilinear Algebra, 2021, 69(11), 2059-2077.
• Gürdal, M., Başaran, H., A-Berezin number of operators, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 2022, 48(1), in press.
• Gürdal, V., Başaran, H., Huban, M.B., Further Berezin radius inequalities, Palestine J. Math., to appear, 2022.
• Gürdal, V., Güncan, A.N., Berezin number inequalities via operator convex functions, Electr. J. Math. Analy. Appl., 2022, 10(2), 83-94.
• Haydarbeygi, Z., Amyari, M., Some refinements of the numerical radius inequalities via Young inequality, Kragujevac J. Math., 2021, 45(2), 191-202.
• Huban, M.B., Başaran, H., Gürdal, M., New upper bounds related to the Berezin number inequalities, J. Inequal. Spec. Funct., 2021, 12(3), 1-12.
• Huban, M.B., Başaran, H., Gürdal, M., Some new inequalities via Berezin numbers, Turk. J. Math. Comput. Sci., in press, 2022.
• Izumino, S., Seo, Y., Determinant for positive operators and Specht's theorem, Sci. Math. Soc., 1998, 1(3), 307-310.
• Karaev, M.T., Berezin symbol and invertibility of operators on the functional Hilbert spaces, J. Funct. Anal., 2006, 238, 181-192.
• Karaev, M.T., Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory, 2013, 7, 983-1018.
• Khatib, Y., Hassani, M., Amyari, M., Refinements numerical radius inequalities via Specht's ratio, J. Math. Ext., 2022, 16(7), 1-18.
• Kittaneh, F., Notes on some inequalities for Hilbert space operators, Publ. Res. Ins. Math. Sci. 1988, 24, 283-293.
• Kittaneh, F., A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math., 2003, 158(1), 11-17.
• Kittaneh, F., El-Haddad, M., Numerical radius inequalities for Hilbert space operators II, Studia Math., 2007, 182(2), 133-140.
• Mond, B., Pečarić, J., Convec inequalities in Hilbert space, Houston J. Math., 1993, 46, 221-232.
• Pečarić, J., Furuta, T., Mićić, H., Seo, Y., Mond-Pečarić, Method in Operator Inequalities, Inequalities for Bounded Selfadjoint Operators on Hilbert Space. Monographs in Inequalities, 1. Element, Zagreb, 2005.
• Specht, W., Zur theorie der elementaren Mittel, Math. Z., 1960, 74, 91-98.
• Shebrawi, K., Albadawi, H., Numerical radius and operator norm inequalities, J. Inequal. Appl. Art. ID 492154, 2009, 11 pp.