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Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği

Yıl 2021, Cilt: 9 Sayı: 6 - ICAIAME 2021, 1 - 14, 31.12.2021
https://doi.org/10.29130/dubited.1013082

Öz

Normalleştirilmiş $K_{\lambda}:=\frac{k_{\lambda}}{\left\Vert k_{\lambda}\right\Vert_{\mathcal{H}}}$, üretici çekirdekli $\mathcal{H}\left( \Omega\right) $, Hilbert uzayı üzerinde $A$ sınırlı lineer operatör için Berezin sembolü ve Berezin sayısı sırasıyla $A\left( \lambda\right) :=\left\langle AK_{\lambda},K_{\lambda}\right\rangle _{\mathcal{H}}$ ve $\mathrm{ber}(A):=\sup_{\lambda\in\Omega}\left\vert A{(\lambda)}\right\vert $ biçiminde tanımlanır. Bu karakteristik arasındaki durumlardan $\mathrm{ber}\left( A\right) \leq\frac{1}{\sqrt{2}}\mathrm{ber}\left(\left\vert A\right\vert +i\left\vert A^{\ast}\right\vert \right) $ eşitsizliği elde edilmiştir. Bu çalışmamızda ise onlar arasındaki diğer eşitsizlikler ispatlanmış ve Berezin sayı eşitsizlikleri için operatör konveks fonksiyonlarının bazı uygulamaları verilmiştir.

Kaynakça

  • [1] N. Aronszajn, “Theory of reproducing kernels,” Transactions of The American Mathematical Society, vol. 68, pp. 337-404, 1950.
  • [2] M. Bakherad and M.T. Garayev, “Berezin number inequalities for operators,” Concrete Operators, vol. 6, no. 1, pp. 33-43, 2019.
  • [3] H. Başaran, M. Gürdal and A. N. Güncan, “Some operator inequalities associated with Kantorovich and Hölder-McCarthy inequalities and their applications,” Turkish Journal of Mathematics, vol. 43, no. 1, pp. 523-532, 2019.
  • [4] H. Başaran, M. B. Huban and M. Gürdal, “Inequalities related to Berezin norm and Berezin number of operators,” preprint, 2021.
  • [5] F. A. Berezin, “Covariant and contravariant symbols for operators,” Mathematics of the USSR-Izvestiya, vol. 6, pp. 1117-1151, 1972.
  • [6] S. S. Dragomir, “Hermite-Hadamard 's type inequalities for operator convex functions,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 766-772, 2011.
  • [7] M. El-Haddad and F. Kittaneh, “Numerical radius inequalities for Hilbert space operators (II),” Studia Mathematica, vol. 182, no. 2, pp. 133-140, 2007.
  • [8] T. Furuta, “A simplified proof of Heinz inequality and scrutiny of its equality,” American Mathematical Society, vol. 97, no. 4, pp. 751-753, 1986.
  • [9] M. T. Garayev, “Berezin symbols, Hölder-McCarthy and Young inequalities and their applications,” Proceedings of Institude of Mathematics and Mechanics. National Academy of Sciences of Azerbaijan, vol. 43, no. 2, pp. 287-295, 2017.
  • [10] M. Garayev, F. Bouzeffour, M. Gürdal and C. M. Yangöz, “Refinements of Kantorovich type, Schwarz and Berezin number inequalities,” Extracta Mathematicae, vol. 35, pp. 1-20, 2020.
  • [11] M. T. Garayev, M. Gürdal and A. Okudan, “Hardy-Hilbert's inequality and a power inequality for Berezin numbers for operators,” Mathematical Inequalities and Applications, vol. 19, pp. 883-891, 2016.
  • [12] M. T. Garayev, M. Gürdal and S. Saltan, “Hardy type inequaltiy for reproducing kernel Hilbert space operators and related problems,” Positivity, vol. 21, pp. 1615-1623, 2017.
  • [13] M. T. Garayev, H. Guedri, M. Gürdal and G.M. Alsahli, “On some problems for operators on the reproducing kernel Hilbert space,” Linear Multilinear Algebra, vol. 69, no. 11, pp. 2059-2077, 2021.
  • [14] M. Garayev, S. Saltan, F. Bouzeffour and B. Aktan, “Some inequalities involving Berezin symbols of operator means and related questions,” Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales Serie A: Matematicas RACSAM, vol. 114, no. 85, pp. 1-17, 2020.
  • [15] K. E. Gustafson and D. K. M. Rao, Numerical Range, New York, USA: Springer-Verlag, 1997.
  • [16] M. Hajmohamadi, R. Lashkaripour and M. Bakherad, “Improvements of Berezin number inequalities,” Linear Multilinear Algebra, vol. 68, no. 6, pp. 1218-1229, 2020.
  • [17] M. B. Huban, H. Başaran and M. Gürdal, “New upper bounds related to the Berezin number inequalities,” Journal of Inequalities and Special Functions, vol. 12, no. 3, pp. 1-12, 2021.
  • [18] M. T. Karaev, “Berezin set and Berezin number of operators and their applications,” in The 8th Workshop on Numerical Ranges and Numerical Radii (WONRA -06), Bremen, Germany, University of Bremen, July 2006, pp. 14.
  • [19] M. T. Karaev, “Berezin symbol and invertibility of operators on the functional Hilbert spaces,” Journal of Functional Analysis, vol. 238, pp. 181-192, 2006.
  • [20] M. T. Karaev, “Reproducing kernels and Berezin symbols techniques in various questions of operator theory,” Complex Analysis and Operator Theory, vol. 7, pp. 983-1018, 2013.
  • [21] F. Kittaneh, “Notes on some inequalities for Hilbert space operators,” Publications of the Research Institude for Mathematical Sciences, vol. 24, pp. 283-293, 1988.
  • [22] F. Kittaneh, “A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix,” Studia Mathematica, vol. 158, no. 1, pp. 11-17, 2003.
  • [23] F. Kittaneh, “Numerical radius inequalities for Hilbert space operators,” Studia Mathematica, vol. 168, no. 1, pp. 73-80, 2005.
  • [24] B. Mond and J. Pečarić, “On Jensen's inequality for operator convex functions,” Houston Journal of Mathematics, vol. 21, pp. 739-753, 1995.
  • [25] H. R. Moradi and M. Sabahheh, “More accurate numerical radius ineinequalities (II),” Linear and Multilinear Algebra, vol. 69, no. 5, pp. 921-933, 2021.
  • [26] M. Sababheh, “Convexity and matrix means,” Linear Algebra Applications, vol. 506, pp. 588-602, 2016.
  • [27] M. Sababheh, “Numerical radius inequalities via convexity,” Linear Algebra Applications, vol. 549, pp. 67-78, 2018.
  • [28] M. Sababheh and H. R. Moradi, “More accurate numerical radius inequalities (I),” Linear and Multilinear Algebra, vol. 69, no. 10, pp. 1964-1973, 2021.
  • [29] S. S. Sahoo, N. Das and D. Mishra, “Berezin number and numerical radius inequalities for operators on Hilbert spaces,” Advances in Operator Theory, vol. 5, pp. 714-727, 2020.
  • [30] R. Tapdigoglu, “New Berezin symbol inequalities for operators on the reproducing kernel Hilbert space,” Operators and Matrices, vol. 15, no. 3, pp. 1031-1043, 2021.
  • [31] U. Yamancı and M. Gürdal, “On numerical radius and Berezin number inequalities for reproducing kernel Hilbert space,” New York Journal of Mathematics, vol. 23, pp. 1531-1537, 2017.
  • [32] U. Yamancı, M. Gürdal and M. T. Garayev, “Berezin number inequality for convex function in reproducing kernel Hilbert space,” Filomat, vol. 31, pp. 5711-5717, 2017.
  • [33] U. Yamancı, R. Tunç and M. Gürdal, “Berezin numbers, Grüss type inequalities and their applications,” Bulletin Malaysian Mathematical Sciences Society, vol. 43, pp. 2287-2296, 2020.

Berezin Number Inequality for Increasing Operator Convex Function

Yıl 2021, Cilt: 9 Sayı: 6 - ICAIAME 2021, 1 - 14, 31.12.2021
https://doi.org/10.29130/dubited.1013082

Öz

Kaynakça

  • [1] N. Aronszajn, “Theory of reproducing kernels,” Transactions of The American Mathematical Society, vol. 68, pp. 337-404, 1950.
  • [2] M. Bakherad and M.T. Garayev, “Berezin number inequalities for operators,” Concrete Operators, vol. 6, no. 1, pp. 33-43, 2019.
  • [3] H. Başaran, M. Gürdal and A. N. Güncan, “Some operator inequalities associated with Kantorovich and Hölder-McCarthy inequalities and their applications,” Turkish Journal of Mathematics, vol. 43, no. 1, pp. 523-532, 2019.
  • [4] H. Başaran, M. B. Huban and M. Gürdal, “Inequalities related to Berezin norm and Berezin number of operators,” preprint, 2021.
  • [5] F. A. Berezin, “Covariant and contravariant symbols for operators,” Mathematics of the USSR-Izvestiya, vol. 6, pp. 1117-1151, 1972.
  • [6] S. S. Dragomir, “Hermite-Hadamard 's type inequalities for operator convex functions,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 766-772, 2011.
  • [7] M. El-Haddad and F. Kittaneh, “Numerical radius inequalities for Hilbert space operators (II),” Studia Mathematica, vol. 182, no. 2, pp. 133-140, 2007.
  • [8] T. Furuta, “A simplified proof of Heinz inequality and scrutiny of its equality,” American Mathematical Society, vol. 97, no. 4, pp. 751-753, 1986.
  • [9] M. T. Garayev, “Berezin symbols, Hölder-McCarthy and Young inequalities and their applications,” Proceedings of Institude of Mathematics and Mechanics. National Academy of Sciences of Azerbaijan, vol. 43, no. 2, pp. 287-295, 2017.
  • [10] M. Garayev, F. Bouzeffour, M. Gürdal and C. M. Yangöz, “Refinements of Kantorovich type, Schwarz and Berezin number inequalities,” Extracta Mathematicae, vol. 35, pp. 1-20, 2020.
  • [11] M. T. Garayev, M. Gürdal and A. Okudan, “Hardy-Hilbert's inequality and a power inequality for Berezin numbers for operators,” Mathematical Inequalities and Applications, vol. 19, pp. 883-891, 2016.
  • [12] M. T. Garayev, M. Gürdal and S. Saltan, “Hardy type inequaltiy for reproducing kernel Hilbert space operators and related problems,” Positivity, vol. 21, pp. 1615-1623, 2017.
  • [13] M. T. Garayev, H. Guedri, M. Gürdal and G.M. Alsahli, “On some problems for operators on the reproducing kernel Hilbert space,” Linear Multilinear Algebra, vol. 69, no. 11, pp. 2059-2077, 2021.
  • [14] M. Garayev, S. Saltan, F. Bouzeffour and B. Aktan, “Some inequalities involving Berezin symbols of operator means and related questions,” Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales Serie A: Matematicas RACSAM, vol. 114, no. 85, pp. 1-17, 2020.
  • [15] K. E. Gustafson and D. K. M. Rao, Numerical Range, New York, USA: Springer-Verlag, 1997.
  • [16] M. Hajmohamadi, R. Lashkaripour and M. Bakherad, “Improvements of Berezin number inequalities,” Linear Multilinear Algebra, vol. 68, no. 6, pp. 1218-1229, 2020.
  • [17] M. B. Huban, H. Başaran and M. Gürdal, “New upper bounds related to the Berezin number inequalities,” Journal of Inequalities and Special Functions, vol. 12, no. 3, pp. 1-12, 2021.
  • [18] M. T. Karaev, “Berezin set and Berezin number of operators and their applications,” in The 8th Workshop on Numerical Ranges and Numerical Radii (WONRA -06), Bremen, Germany, University of Bremen, July 2006, pp. 14.
  • [19] M. T. Karaev, “Berezin symbol and invertibility of operators on the functional Hilbert spaces,” Journal of Functional Analysis, vol. 238, pp. 181-192, 2006.
  • [20] M. T. Karaev, “Reproducing kernels and Berezin symbols techniques in various questions of operator theory,” Complex Analysis and Operator Theory, vol. 7, pp. 983-1018, 2013.
  • [21] F. Kittaneh, “Notes on some inequalities for Hilbert space operators,” Publications of the Research Institude for Mathematical Sciences, vol. 24, pp. 283-293, 1988.
  • [22] F. Kittaneh, “A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix,” Studia Mathematica, vol. 158, no. 1, pp. 11-17, 2003.
  • [23] F. Kittaneh, “Numerical radius inequalities for Hilbert space operators,” Studia Mathematica, vol. 168, no. 1, pp. 73-80, 2005.
  • [24] B. Mond and J. Pečarić, “On Jensen's inequality for operator convex functions,” Houston Journal of Mathematics, vol. 21, pp. 739-753, 1995.
  • [25] H. R. Moradi and M. Sabahheh, “More accurate numerical radius ineinequalities (II),” Linear and Multilinear Algebra, vol. 69, no. 5, pp. 921-933, 2021.
  • [26] M. Sababheh, “Convexity and matrix means,” Linear Algebra Applications, vol. 506, pp. 588-602, 2016.
  • [27] M. Sababheh, “Numerical radius inequalities via convexity,” Linear Algebra Applications, vol. 549, pp. 67-78, 2018.
  • [28] M. Sababheh and H. R. Moradi, “More accurate numerical radius inequalities (I),” Linear and Multilinear Algebra, vol. 69, no. 10, pp. 1964-1973, 2021.
  • [29] S. S. Sahoo, N. Das and D. Mishra, “Berezin number and numerical radius inequalities for operators on Hilbert spaces,” Advances in Operator Theory, vol. 5, pp. 714-727, 2020.
  • [30] R. Tapdigoglu, “New Berezin symbol inequalities for operators on the reproducing kernel Hilbert space,” Operators and Matrices, vol. 15, no. 3, pp. 1031-1043, 2021.
  • [31] U. Yamancı and M. Gürdal, “On numerical radius and Berezin number inequalities for reproducing kernel Hilbert space,” New York Journal of Mathematics, vol. 23, pp. 1531-1537, 2017.
  • [32] U. Yamancı, M. Gürdal and M. T. Garayev, “Berezin number inequality for convex function in reproducing kernel Hilbert space,” Filomat, vol. 31, pp. 5711-5717, 2017.
  • [33] U. Yamancı, R. Tunç and M. Gürdal, “Berezin numbers, Grüss type inequalities and their applications,” Bulletin Malaysian Mathematical Sciences Society, vol. 43, pp. 2287-2296, 2020.
Toplam 33 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Mualla Birgül Huban 0000-0003-2710-8487

Hamdullah Başaran Bu kişi benim 0000-0002-9864-9515

Mehmet Gürdal 0000-0003-0866-1869

Yayımlanma Tarihi 31 Aralık 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 9 Sayı: 6 - ICAIAME 2021

Kaynak Göster

APA Huban, M. B., Başaran, H., & Gürdal, M. (2021). Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği. Düzce Üniversitesi Bilim Ve Teknoloji Dergisi, 9(6), 1-14. https://doi.org/10.29130/dubited.1013082
AMA Huban MB, Başaran H, Gürdal M. Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği. DÜBİTED. Aralık 2021;9(6):1-14. doi:10.29130/dubited.1013082
Chicago Huban, Mualla Birgül, Hamdullah Başaran, ve Mehmet Gürdal. “Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği”. Düzce Üniversitesi Bilim Ve Teknoloji Dergisi 9, sy. 6 (Aralık 2021): 1-14. https://doi.org/10.29130/dubited.1013082.
EndNote Huban MB, Başaran H, Gürdal M (01 Aralık 2021) Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği. Düzce Üniversitesi Bilim ve Teknoloji Dergisi 9 6 1–14.
IEEE M. B. Huban, H. Başaran, ve M. Gürdal, “Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği”, DÜBİTED, c. 9, sy. 6, ss. 1–14, 2021, doi: 10.29130/dubited.1013082.
ISNAD Huban, Mualla Birgül vd. “Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği”. Düzce Üniversitesi Bilim ve Teknoloji Dergisi 9/6 (Aralık 2021), 1-14. https://doi.org/10.29130/dubited.1013082.
JAMA Huban MB, Başaran H, Gürdal M. Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği. DÜBİTED. 2021;9:1–14.
MLA Huban, Mualla Birgül vd. “Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği”. Düzce Üniversitesi Bilim Ve Teknoloji Dergisi, c. 9, sy. 6, 2021, ss. 1-14, doi:10.29130/dubited.1013082.
Vancouver Huban MB, Başaran H, Gürdal M. Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği. DÜBİTED. 2021;9(6):1-14.

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