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Some New Inequalities via Berezin Numbers

Year 2022, , 129 - 137, 30.06.2022
https://doi.org/10.47000/tjmcs.1014841

Abstract

The Berezin transform $\widetilde{T}$ and the Berezin radius of an operator
$T$ on the reproducing kernel Hilbert space $\mathcal{H}\left( Q\right) $
over some set $Q$ with the reproducing kernel $K_{\eta}$ are defined,
respectively, by
\[
\widetilde{T}(\eta)=\left\langle {T\frac{K_{\eta}}{{\left\Vert K_{\eta
}\right\Vert }},\frac{K_{\eta}}{{\left\Vert K_{\eta}\right\Vert }}%
}\right\rangle ,\ \eta\in Q\text{ and }\mathrm{ber}(T):=\sup_{\eta\in
Q}\left\vert \widetilde{T}{(\eta)}\right\vert .
\]
We study several sharp inequalities by using this bounded function
$\widetilde{T},$ involving powers of the Berezin radius and the Berezin norms
of reproducing kernel Hilbert space operators. We also give some inequalities
regarding the Berezin transforms of sum of two operators.

References

  • Bakherad, M., Garayev, M.T., Berezin number inequalities for operators, Concr. Oper., 6(1)(2019), 33-43.
  • Başaran, H. Gürdal, M., Berezin number inequalities via Young inequality, Honam Mathematical J., 43(3)(2021), 523-537.
  • Başaran, H., Gürdal, M., Güncan, A.N., Some operator inequalities associated with Kantorovich and Hölder-McCarthy inequalities and their applications, Turkish J. Math., 43(1)(2019), 523-532.
  • Berezin, F.A., Covariant and contravariant symbols for operators, Izv. Akad. Nauk SSSR Ser. Mat., 36(1972), 1134-1167.
  • Bhatia, R., Kittaneh, F., Norm inequalities for positive operators, Left. Math. Phys., 43(1998), 225-231.
  • Dragomir, S.S., Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces, Linear Algebra Appl., 419(1)(2006), 256-264.
  • Dragomir, S. S., Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces, Springer Briefs in Mathematics, Springer, Cham, Switzerland, 2013.
  • El-Haddad, M., Kittaneh, F., Numerical radius inequalities for Hilbert space operators (II), Studia Math., 182(2)(2007), 133-140.
  • Furuta, T., Norm inequalities equivalent to Löwner-Heinz theorem, Rev. Math. Phys., 1(1989), 135-137
  • Garayev, M., Bouzeffour, F., Gürdal, M., Yangöz, C.M., Refinements of Kantorovich type, Schwarz and Berezin number inequalities, Extracta Math., 35(2020), 1-20.
  • Garayev, M.T., Gürdal, M., Okudan, A., Hardy-Hilbert's inequality and a power inequality for Berezin numbers for operators, Math. Inequal. Appl., 19(3)(2016), 883-891.
  • Garayev, M.T., Gürdal, M., Saltan, S., Hardy type inequaltiy for reproducing kernel Hilbert space operators and related problems, Positivity, 21(4)(2017), 1615-1623.
  • Garayev, M.T., Gürdal, M., Yamanci, U., Alsahli, G.M., Boundary behavior of Berezin symbols and related results, Filomat, 33(14)(2019), 4433-4439
  • 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)(2021), 2059-2077.
  • Garayev, M., Saltan, S., Bouzeffour, F., Aktan, B., Some inequalities involving Berezin symbols of operator means and related questions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 114(85)(2020), 1-17.
  • Gürdal, M., Duman, N., Berezin number inequalities involving superquadratic functions, Casp. J. Math. Sci., 10(1)(2021), 18-27
  • Gustafson, K.E., Rao, D.K.M., Numerical Range, Springer-Verlag, New York, 1997.
  • Halmos, P. R., A Hilbert Space Problem Book , 2nd ed., Springer, New York, 1982.
  • Hardy, G.H., Littlewood, J.E., Polya, G., Inequalities. 2nd ed., Cambridge Univ. Press, Cambridge, 1988.
  • Huban, M.B., Başaran, H., Gürdal, M., New upper bounds related to the Berezin number inequalities, J. Inequal. Spec. Funct., 12(3)(2021), 1-12.
  • Huban, M.B., Gürdal, M., Tilki, H., Some classical inequalities and their applications, Filomat, 35(7)(2021), 2165-2173.
  • Karaev M.T., 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.
  • Karaev, M.T., Berezin symbol and invertibility of operators on the functional Hilbert spaces, J. Funct. Anal., 238(2006), 181-192.
  • Karaev, M.T., Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory, 7(2013), 983-1018.
  • Kittaneh, F., Notes on some inequalities for Hilbert space operators, Publ. Res. Ins. Math. Sci., 24(1988), 283-293.
  • Kittaneh, F., Singular values of companion matrices and bounds on zeros of polynomials, SIAM J. Matrix Anal. Appl., 16(1995), 333-340.
  • Kittaneh, F., Norm inequalities for sums of positive operators, J. Operator Theory, 48(2002), 95-103.
  • Kittaneh, F., A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math., 158(1)(2003), 11-17.
  • Kittaneh, F., Numerical radius inequalities for Hilbert space operators, Studia Math., 168(1)(2005), 73-80.
  • Kittaneh, F., Manasrah, Y., Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl., 361(2010), 262-269.
  • Kittaneh, F., Moslehian, M.S., Yamazaki, T., Cartesian decomposition and numerical radius inequalities, Lin. Algeb. Appl., 471(2015), 46-53.
  • Tapdigoglu, R., New Berezin symbol inequalities for operators on the reproducing kernel Hilbert space, Oper. Matrices, 15(3)(2021), 1031-1043.
  • Tapdigoglu, R., Gürdal, M., Altwaijry, N., Sarı, N., Davis-Wielandt-Berezin radius inequalities via Dragomir inequalities, Oper. Matrices, 15(4)(2021), 1445-1460.
  • Yamancı, U., Gürdal, M., On numerical radius and Berezin number inequalities for reproducing kernel Hilbert space, New York J. Math., 23(2017), 1531-1537.
  • Yamancı, U., Gürdal, M., Garayev, M.T., Berezin number inequality for convex function in reproducing kernel Hilbert space, Filomat, 31(2017), 5711-5717.
  • Yamancı, U., Tunç, R., Gürdal, M., Berezin numbers, Grüss type inequalities and their applications, Bull. Malays. Math. Sci. Soc., 43(3)(2020), 2287-2296.
Year 2022, , 129 - 137, 30.06.2022
https://doi.org/10.47000/tjmcs.1014841

Abstract

References

  • Bakherad, M., Garayev, M.T., Berezin number inequalities for operators, Concr. Oper., 6(1)(2019), 33-43.
  • Başaran, H. Gürdal, M., Berezin number inequalities via Young inequality, Honam Mathematical J., 43(3)(2021), 523-537.
  • Başaran, H., Gürdal, M., Güncan, A.N., Some operator inequalities associated with Kantorovich and Hölder-McCarthy inequalities and their applications, Turkish J. Math., 43(1)(2019), 523-532.
  • Berezin, F.A., Covariant and contravariant symbols for operators, Izv. Akad. Nauk SSSR Ser. Mat., 36(1972), 1134-1167.
  • Bhatia, R., Kittaneh, F., Norm inequalities for positive operators, Left. Math. Phys., 43(1998), 225-231.
  • Dragomir, S.S., Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces, Linear Algebra Appl., 419(1)(2006), 256-264.
  • Dragomir, S. S., Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces, Springer Briefs in Mathematics, Springer, Cham, Switzerland, 2013.
  • El-Haddad, M., Kittaneh, F., Numerical radius inequalities for Hilbert space operators (II), Studia Math., 182(2)(2007), 133-140.
  • Furuta, T., Norm inequalities equivalent to Löwner-Heinz theorem, Rev. Math. Phys., 1(1989), 135-137
  • Garayev, M., Bouzeffour, F., Gürdal, M., Yangöz, C.M., Refinements of Kantorovich type, Schwarz and Berezin number inequalities, Extracta Math., 35(2020), 1-20.
  • Garayev, M.T., Gürdal, M., Okudan, A., Hardy-Hilbert's inequality and a power inequality for Berezin numbers for operators, Math. Inequal. Appl., 19(3)(2016), 883-891.
  • Garayev, M.T., Gürdal, M., Saltan, S., Hardy type inequaltiy for reproducing kernel Hilbert space operators and related problems, Positivity, 21(4)(2017), 1615-1623.
  • Garayev, M.T., Gürdal, M., Yamanci, U., Alsahli, G.M., Boundary behavior of Berezin symbols and related results, Filomat, 33(14)(2019), 4433-4439
  • 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)(2021), 2059-2077.
  • Garayev, M., Saltan, S., Bouzeffour, F., Aktan, B., Some inequalities involving Berezin symbols of operator means and related questions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 114(85)(2020), 1-17.
  • Gürdal, M., Duman, N., Berezin number inequalities involving superquadratic functions, Casp. J. Math. Sci., 10(1)(2021), 18-27
  • Gustafson, K.E., Rao, D.K.M., Numerical Range, Springer-Verlag, New York, 1997.
  • Halmos, P. R., A Hilbert Space Problem Book , 2nd ed., Springer, New York, 1982.
  • Hardy, G.H., Littlewood, J.E., Polya, G., Inequalities. 2nd ed., Cambridge Univ. Press, Cambridge, 1988.
  • Huban, M.B., Başaran, H., Gürdal, M., New upper bounds related to the Berezin number inequalities, J. Inequal. Spec. Funct., 12(3)(2021), 1-12.
  • Huban, M.B., Gürdal, M., Tilki, H., Some classical inequalities and their applications, Filomat, 35(7)(2021), 2165-2173.
  • Karaev M.T., 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.
  • Karaev, M.T., Berezin symbol and invertibility of operators on the functional Hilbert spaces, J. Funct. Anal., 238(2006), 181-192.
  • Karaev, M.T., Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory, 7(2013), 983-1018.
  • Kittaneh, F., Notes on some inequalities for Hilbert space operators, Publ. Res. Ins. Math. Sci., 24(1988), 283-293.
  • Kittaneh, F., Singular values of companion matrices and bounds on zeros of polynomials, SIAM J. Matrix Anal. Appl., 16(1995), 333-340.
  • Kittaneh, F., Norm inequalities for sums of positive operators, J. Operator Theory, 48(2002), 95-103.
  • Kittaneh, F., A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math., 158(1)(2003), 11-17.
  • Kittaneh, F., Numerical radius inequalities for Hilbert space operators, Studia Math., 168(1)(2005), 73-80.
  • Kittaneh, F., Manasrah, Y., Improved Young and Heinz inequalities for matrices, J. Math. Anal. Appl., 361(2010), 262-269.
  • Kittaneh, F., Moslehian, M.S., Yamazaki, T., Cartesian decomposition and numerical radius inequalities, Lin. Algeb. Appl., 471(2015), 46-53.
  • Tapdigoglu, R., New Berezin symbol inequalities for operators on the reproducing kernel Hilbert space, Oper. Matrices, 15(3)(2021), 1031-1043.
  • Tapdigoglu, R., Gürdal, M., Altwaijry, N., Sarı, N., Davis-Wielandt-Berezin radius inequalities via Dragomir inequalities, Oper. Matrices, 15(4)(2021), 1445-1460.
  • Yamancı, U., Gürdal, M., On numerical radius and Berezin number inequalities for reproducing kernel Hilbert space, New York J. Math., 23(2017), 1531-1537.
  • Yamancı, U., Gürdal, M., Garayev, M.T., Berezin number inequality for convex function in reproducing kernel Hilbert space, Filomat, 31(2017), 5711-5717.
  • Yamancı, U., Tunç, R., Gürdal, M., Berezin numbers, Grüss type inequalities and their applications, Bull. Malays. Math. Sci. Soc., 43(3)(2020), 2287-2296.
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mualla Birgül Huban 0000-0003-2710-8487

Hamdullah Başaran 0000-0002-9864-9515

Mehmet Gürdal 0000-0003-0866-1869

Publication Date June 30, 2022
Published in Issue Year 2022

Cite

APA Huban, M. B., Başaran, H., & Gürdal, M. (2022). Some New Inequalities via Berezin Numbers. Turkish Journal of Mathematics and Computer Science, 14(1), 129-137. https://doi.org/10.47000/tjmcs.1014841
AMA Huban MB, Başaran H, Gürdal M. Some New Inequalities via Berezin Numbers. TJMCS. June 2022;14(1):129-137. doi:10.47000/tjmcs.1014841
Chicago Huban, Mualla Birgül, Hamdullah Başaran, and Mehmet Gürdal. “Some New Inequalities via Berezin Numbers”. Turkish Journal of Mathematics and Computer Science 14, no. 1 (June 2022): 129-37. https://doi.org/10.47000/tjmcs.1014841.
EndNote Huban MB, Başaran H, Gürdal M (June 1, 2022) Some New Inequalities via Berezin Numbers. Turkish Journal of Mathematics and Computer Science 14 1 129–137.
IEEE M. B. Huban, H. Başaran, and M. Gürdal, “Some New Inequalities via Berezin Numbers”, TJMCS, vol. 14, no. 1, pp. 129–137, 2022, doi: 10.47000/tjmcs.1014841.
ISNAD Huban, Mualla Birgül et al. “Some New Inequalities via Berezin Numbers”. Turkish Journal of Mathematics and Computer Science 14/1 (June 2022), 129-137. https://doi.org/10.47000/tjmcs.1014841.
JAMA Huban MB, Başaran H, Gürdal M. Some New Inequalities via Berezin Numbers. TJMCS. 2022;14:129–137.
MLA Huban, Mualla Birgül et al. “Some New Inequalities via Berezin Numbers”. Turkish Journal of Mathematics and Computer Science, vol. 14, no. 1, 2022, pp. 129-37, doi:10.47000/tjmcs.1014841.
Vancouver Huban MB, Başaran H, Gürdal M. Some New Inequalities via Berezin Numbers. TJMCS. 2022;14(1):129-37.