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Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği

Year 2021, Volume: 9 Issue: 6 - ICAIAME 2021, 1 - 14, 31.12.2021

Mualla Birgül Huban Hamdullah Başaran Mehmet Gürdal

https://doi.org/10.29130/dubited.1013082
Cited By: 6

Abstract

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.

Keywords

Üretici çekirdekli Hilbert uzayı Berezin sembolü Berezin sayısı Hermite-Hadamard eşitsizliği Operatör konveksliği

References

  • [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

Year 2021, Volume: 9 Issue: 6 - ICAIAME 2021, 1 - 14, 31.12.2021

Mualla Birgül Huban Hamdullah Başaran Mehmet Gürdal

https://doi.org/10.29130/dubited.1013082
Cited By: 6

Abstract

Keywords

Reproducing kernel Hilbert space Berezin symbol Berezin number Hermite-Hadamard inequality Operator convexity

References

  • [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.
There are 33 citations in total.

Details

Primary Language Turkish
Subjects Engineering
Journal Section Articles
Authors

Mualla Birgül Huban 0000-0003-2710-8487

Hamdullah Başaran This is me 0000-0002-9864-9515

Mehmet Gürdal 0000-0003-0866-1869

Publication Date December 31, 2021
Published in Issue Year 2021 Volume: 9 Issue: 6 - ICAIAME 2021

Cite

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. DUBİTED. December 2021;9(6):1-14. doi:10.29130/dubited.1013082
Chicago Huban, Mualla Birgül, Hamdullah Başaran, and Mehmet Gürdal. “Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği”. Düzce Üniversitesi Bilim Ve Teknoloji Dergisi 9, no. 6 (December 2021): 1-14. https://doi.org/10.29130/dubited.1013082.
EndNote Huban MB, Başaran H, Gürdal M (December 1, 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, and M. Gürdal, “Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği”, DUBİTED, vol. 9, no. 6, pp. 1–14, 2021, doi: 10.29130/dubited.1013082.
ISNAD Huban, Mualla Birgül et al. “Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği”. Düzce Üniversitesi Bilim ve Teknoloji Dergisi 9/6 (December 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. DUBİTED. 2021;9:1–14.
MLA Huban, Mualla Birgül et al. “Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği”. Düzce Üniversitesi Bilim Ve Teknoloji Dergisi, vol. 9, no. 6, 2021, pp. 1-14, doi:10.29130/dubited.1013082.
Vancouver Huban MB, Başaran H, Gürdal M. Artan Operatör Konveks Fonksiyon İçin Berezin Sayı Eşitsizliği. DUBİTED. 2021;9(6):1-14.

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