New inequalities via Berezin symbols and related questions
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Abstract
The Berezin symbol $\tilde{A}$ of an operator $A$ on the reproducing kernel Hilbert space $\mathcal{H}\left( \Omega\right) $ over some set $\Omega$ with the reproducing kernel $k_{\lambda}$ is defined by
$$ \tilde{A}(\lambda)=\left\langle {A\frac{{k_{\lambda}}}{{\left\Vert {k_{\lambda}}\right\Vert }},\frac{{k_{\lambda}}}{{\left\Vert {k_{\lambda} }\right\Vert }}}\right\rangle ,\ \lambda\in\Omega. $$
The Berezin number of an operator $A$ is defined by
$$ ber(A):=\sup_{\lambda\in\Omega}\left\vert {\tilde{A}(\lambda)}\right\vert . $$
We study some problems of operator theory by using this bounded function $\tilde{A}$, 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 $ber\left( {AB}\right) \leq ber\left( A\right) ber\left( B\right) ,$ for some operators $A$ and $B$ on $\mathcal{H}\left( \Omega\right) $. Moreover, we give in terms of the Berezin number a necessary condition for hyponormality of some operators.
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References
[1] N. Aronzajn, Theory of reproducing kernels, Trans. Amer. Math.Soc. 68 (1950), 337–404. Google Scholar
[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. Google Scholar
[3] F.A. Berezin, Covariant and contravariant symbols for operators, Math. USSR-Izv. 6 (1972), 1117–1151. Google Scholar
[4] F.A. Berezin, Quantization, Math. USSR-Izv. 8 (1974), 1109–1163. Google Scholar
[5] S. Bergman, The kernel function and conformal mapping, Mathematical Surveys and Monographs 5, Amer. Math. Soc., providence, RI (1950). Google Scholar
[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. Google Scholar
[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. Google Scholar
[8] M. Engliˇs, Toeplitz operators and the Berezin transform on H2, Linear Algebra Appl. 223/224 (1995), 171–204. Google Scholar
[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). Google Scholar
[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. Google Scholar
[11] M.T. Garayev, M. Gu ̈rdal and A. Okudan, Hardy-Hilberts inequality and power inequalities for Berezin numbers of operators, Math. Inequal. Appl. 19 (2016), 883–891. Google Scholar
[12] M.T. Garayev, M. Gu ̈rdal and S. Saltan, Hardy type inequality for reproducing kernel Hilbert space operators and related problems, Positivity 21 (2017), 1615–1623. Google Scholar
[13] P.R. Halmos, A Hilbert Space problem Book, Springer- Verlag, (1982). Google Scholar
[14] G. Hardy, J.E. Littlewood and G. Polya, Inequalities, 2 nd ed., Cambridge Univ. Press, Cambridge, (1988). Google Scholar
[15] O. Hirzallah and F. Kittaneh, Matrix Young inequalities for the Hilbert-Schmidt norm, Linear Algebra Appl. 308 (2000), 77–84. Google Scholar
[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. Google Scholar
[17] M.T. Karaev, Berezin symbol and invertibility of operators on the functional Hilbert space, J. Funct. Analysis 238 (2006), 181–192. Google Scholar
[18] M.T. Karaev, Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory 7 (2013), 983–1018. Google Scholar
[19] K. Khatri and V.N. Mishra, Generalized Sz ́asz-Mirakyan operators involving Brenke type polynomials, Appl. Math. Comput. 324 (2018), 228–238. Google Scholar
[20] M. Kian, Hardy-Hilbert type inequalities for Hilbert space operators, Ann. Funct. Anal. 3 (2012), 128–134. Google Scholar
[21] F. Kittaneh, Notes on some inequalities for Hilbert space operators, Publ. Res. Inst. Math. Sci. 24 (2) (1988), 283–293. Google Scholar
[22] F. Kittaneh, Norm inequalities for fractional powers of positive operators, Lett. Math. Phys. 27 (1993), 279–285. Google Scholar
[23] F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl. 361 (2010), 262–269. Google Scholar
[24] V.A. Malyshev, The Bergman kernel and Green function, Zap.Nauch. Semin. POMI (1995), 145–166. Google Scholar
[25] Y. Manasrah and F. Kittaneh, A generalization of two refined Young Inequalities, Positivity 19 (2015), 757–768. Google Scholar
[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). Google Scholar
[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. Google Scholar
[28] S. Saitoh and Y. Sawano, Theory of reproducing kernels and applications, Developments in Mathematics, Springer, Singapore, 44 (2016). https://link.springer.com/book/10.1007/978-981-10-0530-5 Google Scholar
[29] D. Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1 (2007), 491– 526. Google Scholar
[30] D. Sarason, Unbounded Toeplitz operators, Integral Equat. Oper. Theory 61 (2008), 281–298. Google Scholar
[31] R. Tapdigoglu, New Berezin symbol inequalities for operators on the reproducing kernel Hilbert space, Oper. Matrices 15 (3) (2021), 1031–1043. Google Scholar
[32] K. Zhu, Operator Theory in function spaces, Second edition, Mathematical Surveys and Monographs 138, Amer. Math. Soc., providence, RI, (2007). Google Scholar