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Inequalities Involving Berezin Norm and Berezin Number

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Abstract

We obtain new inequalities involving Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space \({\mathscr {H}}.\) Among many inequalities obtained here, it is shown that if A is a positive bounded linear operator on \({\mathscr {H}}\), then \(\Vert A\Vert _{ber}={{{\textbf {ber}}}}(A)\), where \(\Vert A\Vert _{ber}\) and \({{{\textbf {ber}}}}(A)\) are the Berezin norm and Berezin number of A, respectively. In contrast to the numerical radius, this equality does not hold for selfadjoint operators, which highlights the necessity of studying Berezin number inequalities independently.

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References

  1. Bakherad, M., Hajmohamadi, M., Lashkaripour, R., Sahoo, S.: Some extensions of Berezin number inequalities on operators. Rocky Mt. J. Math. 51(6), 1941–1951 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bakherad, M., Yamanci, U.: New estimations for the Berezin number inequality. J. Inequal. Appl. 2020(1), 1–9 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bakherad, M., Garayev, M.T.: Berezin number inequalities for operators. Concr. Oper. 6(1), 33–43 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berezin, F.A.: Quantization. Math. USSR-Izv. 8, 1109–1163 (1974)

    Article  MATH  Google Scholar 

  5. Berezin, F.A.: Covariant and contravariant symbols for operators. Math. USSR-Izv. 6, 1117–1151 (1972)

    Article  MATH  Google Scholar 

  6. Bhunia, P., Paul, K.: Proper improvement of well-known numerical radius inequalities and their applications. Results Math. 76(4), 1–12 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhunia, P., Paul, K.: Development of inequalities and characterization of equality conditions for the numerical radius. Linear Algebra Appl. 630, 306–315 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bhunia, P., Paul, K.: New upper bounds for the numerical radius of Hilbert space operators. Bull. Sci. Math. 167, 102959 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bhunia, P., Paul, K.: Furtherance of numerical radius inequalities of Hilbert space operators. Arch. Math. (Basel) 117(5), 537–546 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cortes, C., Mohri, M., Rostamizadeh, A.: Learning non-linear combinations of kernels. In: Advances in Neural Information Processing Systems, pp. 396–404. Curran Associates Inc. (2009)

  11. De Vito, E., Mücke, N., Rosasco, L.: Reproducing kernel Hilbert spaces on manifolds: Sobolev and diffusion spaces. Anal. Appl. (Singap.) 19(3), 363–396 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dragomir, S.S.: Power inequalities for the numerical radius of a product of two operators in Hilbert spaces. Sarajevo J. Math. 5(18)(2), 269–278 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Englis, M.: Berezin and Berezin–Toeplitz quantizations for general function spaces. Rev. Math. Comput. 19, 385–430 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Fricain, E.: Uniqueness theorems for analytic vector-valued functions. J. Math. Sci. (N. Y.) 101, 3193–3210 (2000)

    Article  MathSciNet  Google Scholar 

  15. Furuta, T.: A simplified proof of Heinz inequality and scrutiny of its equality. Proc. Am. Math. Soc. 97, 751–753 (1986)

    MathSciNet  MATH  Google Scholar 

  16. Garayev, M.T., Guedri, H., Gürdal, M., Alsahli, G.M.: On some problems for operators on the reproducing kernel Hilbert space. Linear Multilinear Algebra 69(11), 2059–2077 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Garayev, M.T., Alomari, M.W.: Inequalities for the Berezin number of operators and related questions. Complex Anal. Oper. Theory 15(2), 1–30 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Garayev, M.T., Yamanci, U.: Čebyšev’s type inequalities and power inequalities for the Berezin number of operators. Filomat 33(8), 2307–2316 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Garayev, M., Gürdal, M., Okudan, A.: Hardy Hilbert s inequality and a power inequality for Berezin numbers for operators. Math. Inequal. Appl. 19(3), 883–891 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Guo, X., Li, L., Wu, Q.: Modeling interactive components by coordinate kernel polynomial models. Math. Found. Comput. 3(4), 263–277 (2020)

    Article  MATH  Google Scholar 

  21. Gürdal, M., Alomari, M.W.: Improvements of some Berezin radius inequalities. Constr. Math. Anal. 5(3), 141–153 (2022)

    MathSciNet  MATH  Google Scholar 

  22. Hajmohamadi, M., Lashkaripour, R., Bakherad, M.: Improvements of Berezin number inequalities. Linear Multilinear Algebra 68(6), 1218–1229 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1988)

    MATH  Google Scholar 

  24. Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  25. Karaev, M.T.: Reproducing kernels and Berezin symbols techniques in various questions of operator theory. Complex Anal. Oper. Theory 7, 983–1018 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Karaev, M.T.: Berezin symbol and invertibility of operators on the functional Hilbert spaces. J. Funct. Anal. 238(1), 181–192 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Karaev, M.T., Saltan, S.: Some results on Berezin symbols. Complex Var. Theory Appl. 50, 185–193 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Lanckriet, G.R.G., Cristianini, N., Bartlett, P., El Ghaoui, L., Jordan, M.I.: Learning the kernel matrix with semi-definite programming. J. Mach. Learn. Res. 5, 27–72 (2003/04)

  29. Paulsen, V.I., Raghupati, M.: An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Cambridge University Press (2016)

  30. Sen, A., Bhunia, P., Paul, K.: Berezin number inequalities of operators on reproducing kernel Hilbert spaces. Rocky Mt. J. Math. 52(3), 1039–1046 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  31. Simon, B.: Trace Ideals and their Applications. Cambridge University Press (1979)

  32. Taghavi, A., Roushan, T.A., Darvish, V.: Some upper bounds for the Berezin number of Hilbert space operators. Filomat 33(14), 4353–4360 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tapdigoglu, R., Gürdal, M., Altwaijry, N., Sarı, N.: Davis–Wielandt–Berezin radius inequalities via Dragomir inequalities. Oper. Matrices 15(4), 1445–1460 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tapdigoglu, R.: New Berezin symbol inequalities for operators on the reproducing kernel Hilbert space. Oper. Matrices 15(3), 1031–1043 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  35. Xu, Y., Zhang, H.: Refinement of reproducing kernels. J. Mach. Learn. Res. 10, 107–140 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Xu, Y., Zhang, H.: Refinable kernels. J. Mach. Learn. Res. 8, 2083–2120 (2007)

    MathSciNet  MATH  Google Scholar 

  37. Yamanci, U., Garayev, M.T.: Some results related to the Berezin number inequalities. Turk. J. Math. 43(4), 1940–1952 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their comments towards an improved final version of the paper. We also thank the handling editor for helping us to write the Application section. Mr. Pintu Bhunia sincerely acknowledges the financial support received from UGC, Govt. of India in the form of Senior Research Fellowship under the mentorship of Prof Kallol Paul. Mr. Anirban Sen would like to thank CSIR, Govt. of India for the financial support in the form of Junior Research Fellowship under the mentorship of Prof Kallol Paul.

Funding

Funding was provided by University Grants Commission (Ref. No.: 1187/(CSIR-UGC NET DEC. 2016) dated 8 NOV 2017) and Council of Scientific and Industrial Research, India (File No.: 09/096(1021)/2020-EMR-I dated 19/10/2020).

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Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, West Bengal, 700032, India

    Pintu Bhunia, Kallol Paul & Anirban Sen

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  1. Pintu Bhunia
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  2. Kallol Paul
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Correspondence to Kallol Paul.

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Communicated by Ding-Xuan Zhou.

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Bhunia, P., Paul, K. & Sen, A. Inequalities Involving Berezin Norm and Berezin Number. Complex Anal. Oper. Theory 17, 7 (2023). https://doi.org/10.1007/s11785-022-01305-9

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