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Hardy type inequaltiy for reproducing Kernel Hilbert space operators and related problems

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Abstract

A Hardy type inequality for Reproducing Kernel Hilbert Space operators is proved. It is well known (see Halmos in A Hilbert space problem book. Springer, Berlin, 1982) the following power inequality for numerical radius of Hilbert space operator A:

$$\begin{aligned} w\left( A^{n}\right) \le \left( w\left( A\right) \right) ^{n} \end{aligned}$$

for any integer \(n>0\). Hovewer, the inverse inequalities

$$\begin{aligned} \left( w\left( A\right) \right) ^{n}\le C\left( w\left( A^{n}\right) \right) , n\ge 2, \end{aligned}$$

for some operator classes with some constant \(C=C\left( n\right) >1\), apparently, are not well investigated in the literature. The same inequalities for the Berezin number of operators on the reproducing Kernel Hilbert space are not studied in general. Here we prove some power inequalities for Berezin number of some operators.

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Acknowledgements

The authors thank the referee for his useful remarks and suggestions which improved the presentation of the paper.

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Author notes

  1. Mubariz T. Garayev

    Present address: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

Authors and Affiliations

  1. Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, B. Vagabzade str. 9, 370141, Baku, Azerbaijan

    Mubariz T. Garayev

  2. Department of Mathematics, Suleyman Demirel University, 32260, Isparta, Turkey

    Mehmet Gürdal & Suna Saltan

Authors
  1. Mubariz T. Garayev
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  2. Mehmet Gürdal
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  3. Suna Saltan
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Corresponding author

Correspondence to Mehmet Gürdal.

Additional information

This paper was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center. Also, the second author is supported by TUBA through Young Scientist Award Program (TUBA-GEBIP/2015).

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Garayev, M.T., Gürdal, M. & Saltan, S. Hardy type inequaltiy for reproducing Kernel Hilbert space operators and related problems. Positivity 21, 1615–1623 (2017). https://doi.org/10.1007/s11117-017-0489-6

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  • DOI: https://doi.org/10.1007/s11117-017-0489-6

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