Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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NEW INEQUALITIES VIA BEREZIN SYMBOLS AND RELATED QUESTIONS

  • Ramiz Tapdigoglu (Department of Mathematics, Azerbaijan State University of Economics (UNEC) and Khazar University) ;
  • Najwa Altwaijry (Department of Mathematics, College of Science, King Saud University) ;
  • Mubariz Garayev (Department of Mathematics, College of Science, King Saud University)
  • Received : 2021.09.22
  • Accepted : 2023.01.09
  • Published : 2023.03.30
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Abstract

The Berezin symbol à of an operator A on the reproducing kernel Hilbert space 𝓗 (Ω) over some set Ω with the reproducing kernel kλ is defined by $${\tilde{A}}(\lambda)=\,\;{\lambda}{\in}{\Omega}$$. The Berezin number of an operator A is defined by $$ber(A):=\sup_{{\lambda}{\in}{\Omega}}{\mid}{\tilde{A}}({\lambda}){\mid}$$. We study some problems of operator theory by using this bounded function Ã, including estimates for Berezin numbers of some operators, including truncated Toeplitz operators. We also prove an operator analog of some Young inequality and use it in proving of some inequalities for Berezin number of operators including the inequality ber (AB) ≤ ber (A) ber (B), for some operators A and B on 𝓗 (Ω). Moreover, we give in terms of the Berezin number a necessary condition for hyponormality of some operators.

Keywords

Acknowledgement

The first and second authors extend their appreciation to the Distinguished Scientist Fellowship Program at King Saud University, Riyadh, Saudi Arabia, for funding this work through Researchers Supporting Project number (RSP2023R187). Also, the third author thanks to Deanship of Scientific Research, College of Science Research Center, King Saud University for supporting this work.

References

  1. N. Aronzajn, Theory of reproducing kernels, Trans. Amer. Math.Soc. 68 (1950), 337-404. https://doi.org/10.1090/S0002-9947-1950-0051437-7
  2. A. Baranov, I. Chalendar, E. Fricain, J. Mashreghi and D. Timotin, Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators, J. Funct. Anal. 259 (2010), 2673-2701. https://doi.org/10.1016/j.jfa.2010.05.005
  3. F.A. Berezin, Covariant and contravariant symbols for operators, Math. USSR-Izv. 6 (1972), 1117-1151. https://doi.org/10.1070/IM1972v006n05ABEH001913
  4. F.A. Berezin, Quantization, Math. USSR-Izv. 8 (1974), 1109-1163. https://doi.org/10.1070/IM1974v008n05ABEH002140
  5. S. Bergman, The kernel function and conformal mapping, Mathematical Surveys and Monographs 5, Amer. Math. Soc., providence, RI (1950).
  6. A. Chandola, R.M. Pandey, R. Agarwal, L. Rathour and V.N. Mishra, On some properties and applications of the generalized m-parameter Mittag-Leffler function, Adv. Math. Models Appl. 7(2) (2022), 130-145.
  7. S.S. Dragomir, A survey of some recent inequalities for the norm and numerical radius of operators in Hilbert spaces, Banach J. Math. Anal. 1 (2007), 154-175. https://doi.org/10.15352/bjma/1240336213
  8. M. Englis, Toeplitz operators and the Berezin transform on H2, Linear Algebra Appl. 223/224(1995), 171-204. https://doi.org/10.1016/0024-3795(94)00056-J
  9. T. Furuta, Invitation to Linear Operators. From Matrices to Bounded Linear Operators on a Hilbert Space, Taylor and Francis, London and New York, (2001).
  10. A.R. Gairola, S. Maindola, L. Rathour, L.N. Mishra and V.N. Mishra, Better uniform approximation by new Bivariate Bernstein operators, J. Anal. Appl. 20, ID: 60, (2022), 1-19. https://doi.org/10.28924/2291-8639-20-2022-60
  11. M.T. Garayev, M. Gurdal and A. Okudan, Hardy-Hilberts inequality and power inequalities for Berezin numbers of operators, Math. Inequal. Appl. 19 (2016), 883-891.
  12. M.T. Garayev, M. Gurdal and S. Saltan, Hardy type inequality for reproducing kernel Hilbert space operators and related problems, Positivity 21 (2017), 1615-1623. https://doi.org/10.1007/s11117-017-0489-6
  13. P.R. Halmos, A Hilbert Space problem Book, Springer- Verlag, (1982).
  14. G. Hardy, J.E. Littlewood and G. Polya, Inequalities, 2 nd ed., Cambridge Univ. Press, Cambridge, (1988).
  15. O. Hirzallah and F. Kittaneh, Matrix Young inequalities for the Hilbert-Schmidt norm, Linear Algebra Appl. 308 (2000), 77-84.
  16. M.T. Karaev, Berezin set and Berezin number of operators and their applications, The 8th Workshop on Numerical Ranges and Numerical Radii (WONRA -06),University of Bremen, July 15-17, (2006), p.14.
  17. M.T. Karaev, Berezin symbol and invertibility of operators on the functional Hilbert space, J. Funct. Analysis 238 (2006), 181-192. https://doi.org/10.1016/j.jfa.2006.04.030
  18. M.T. Karaev, Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory 7 (2013), 983-1018. https://doi.org/10.1007/s11785-012-0232-z
  19. K. Khatri and V.N. Mishra, Generalized Szasz-Mirakyan operators involving Brenke type polynomials, Appl. Math. Comput. 324 (2018), 228-238. https://doi.org/10.1016/j.amc.2017.11.049
  20. M. Kian, Hardy-Hilbert type inequalities for Hilbert space operators, Ann. Funct. Anal. 3 (2012), 128-134. https://doi.org/10.15352/afa/1399899937
  21. F. Kittaneh, Notes on some inequalities for Hilbert space operators, Publ. Res. Inst. Math. Sci. 24 (2) (1988), 283-293. https://doi.org/10.2977/prims/1195175202
  22. F. Kittaneh, Norm inequalities for fractional powers of positive operators, Lett. Math. Phys. 27 (1993), 279-285. https://doi.org/10.1007/BF00777375
  23. F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl. 361 (2010), 262-269. https://doi.org/10.1016/j.jmaa.2009.08.059
  24. V.A. Malyshev, The Bergman kernel and Green function, Zap.Nauch. Semin. POMI (1995), 145-166.
  25. Y. Manasrah and F. Kittaneh, A generalization of two refined Young Inequalities, Positivity 19 (2015), 757-768. https://doi.org/10.1007/s11117-015-0326-8
  26. V.N. Mishra, Some problems on approximations of functions in Banach spaces, Ph. D. Thesis, Indian Institute of Technology, Roorkee 247 667, Uttarakhand, India, (2007).
  27. V.N. Mishra and L.N. Mishra, Trigonometric approximation of signals (Functions) in Lp-norm, Int. J. Contemp. Math. Science 7 (19) (2012), 909-918.
  28. S. Saitoh and Y. Sawano, Theory of reproducing kernels and applications, Developments in Mathematics, Springer, Singapore, 44 (2016).
  29. D. Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1 (2007), 491-526. https://doi.org/10.7153/oam-01-29
  30. D. Sarason, Unbounded Toeplitz operators, Integral Equat. Oper. Theory 61 (2008), 281-298. https://doi.org/10.1007/s00020-008-1588-3
  31. R. Tapdigoglu, New Berezin symbol inequalities for operators on the reproducing kernel Hilbert space, Oper. Matrices 15 (3) (2021), 1031-1043. https://doi.org/10.7153/oam-2021-15-64
  32. K. Zhu, Operator Theory in function spaces, Second edition, Mathematical Surveys and Monographs 138, Amer. Math. Soc., providence, RI, (2007).