# Institute of Mathematics

## Some Berezin number inequalities for operator matrices

###### Received February 3, 2017.   First published February 1, 2018.

Abstract:  The Berezin symbol $\tilde{A}$ of an operator $A$ acting on the reproducing kernel Hilbert space ${\mathcal H}={\mathcal H}(\Omega)$ over some (nonempty) set is defined by $\tilde{A}(\lambda)=\langle A\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle,$ $\lambda\in\Omega$, where $\hat{k}_{\lambda}={k_{\lambda}}/{\|k_{\lambda}\|}$ is the normalized reproducing kernel of ${\mathcal H}$. The Berezin number of the operator $A$ is defined by $ber(A)=\sup_{\lambda\in\Omega}|\tilde{A}(\lambda)|=\sup_{\lambda\in\Omega}|\langle A\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle|$. Moreover, $ber(A)\leq w(A)$ (numerical radius). We present some Berezin number inequalities. Among other inequalities, it is shown that if $T=\left[\smallmatrix A&B C&D \right]\in{\mathbb B}({\mathcal H(\Omega_1)}\oplus{\mathcal H(\Omega_2)})$, then ber( T) \leq\frac12( ber(A)+ ber(D))+\frac12\sqrt{( ber(A)- ber(D))^2+(\|B\|+\|C\|)^2}.
Keywords:  reproducing kernel; Berezin number; numerical radius; operator matrix
Classification MSC:  47A30, 15A60, 30E20, 47A12, 47B15, 47B20
DOI:  10.21136/CMJ.2018.0048-17

PDF available at:  Institute of Mathematics CAS

References:
[1] A. Abu-Omar, F. Kittaneh: Numerical radius inequalities for $n\times n$ operator matrices. Linear Algebra Appl. 468 (2015), 18-26. DOI 10.1016/j.laa.2013.09.049 | MR 3293237 | Zbl 1316.47005
[2] F. A. Berezin: Covariant and contravariant symbols of operators. Math. USSR, Izv. 6(1972) (1973), 1117-1151. (English. Russian original.); translation from Russian Izv. Akad. Nauk SSSR, Ser. Mat. 36 (1972), 1134-1167. DOI 10.1070/IM1972v006n05ABEH001913 | MR 0350504 | Zbl 0259.47004
[3] F. A. Berezin: Quantization. Math. USSR, Izv. 8 (1974), 1109-1165. (English. Russian original.); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38 (1974), 1116-1175. DOI 10.1070/IM1974v008n05ABEH002140 | MR 0395610 | Zbl 0312.53049
[4] K. E. Gustafson, D. K. M. Rao: Numerical Range. The Field of Values of Linear Operators and Matrices. Universitext, Springer, New York (1997). DOI 10.1007/978-1-4613-8498-4 | MR 1417493 | Zbl 0874.47003
[5] M. Hajmohamadi, R. Lashkaripour, M. Bakherad: Some generalizations of numerical radius on off-diagonal part of $2\times 2$ operator matrices. To appear in J. Math. Inequal. Available at ArXiv 1706.05040
[6] P. R. Halmos: A Hilbert Space Problem Book. Graduate Texts in Mathematics 19, Encyclopedia of Mathematics and Its Applications 17, Springer, New York (1982). DOI 10.1007/978-1-4684-9330-6 | MR 0675952 | Zbl 0496.47001
[7] R. A. Horn, C. R. Johnson: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991). DOI 10.1017/CBO9780511840371 | MR 1091716 | Zbl 0729.15001
[8] J. C. Hou, H. K. Du: Norm inequalities of positive operator matrices. Integral Equations Operator Theory 22 (1995), 281-294. DOI 10.1007/BF01378777 | MR 1337376 | Zbl 0839.47004
[9] M. T. Karaev: On the Berezin symbol. J. Math. Sci., New York 115 (2003), 2135-2140. (English. Russian original.); translation from Zap. Nauchn. Semin. POMI 270 (2000), 80-89. DOI 10.1023/A:1022828602917 | MR 1795640 | Zbl 1025.47015
[10] M. T. Karaev: Functional analysis proofs of Abel's theorems. Proc. Am. Math. Soc. 132 (2004), 2327-2329. DOI 10.1090/S0002-9939-04-07354-X | MR 2052409 | Zbl 1099.40003
[11] M. T. Karaev: Berezin symbol and invertibility of operators on the functional Hilbert spaces. J. Funct. Anal. 238 (2006), 181-192. DOI 10.1016/j.jfa.2006.04.030 | MR 2253012 | Zbl 1102.47018
[12] M. T. Karaev, S. Saltan: Some results on Berezin symbols. Complex Variables, Theory Appl. 50 (2005), 185-193. DOI 10.1080/02781070500032861 | MR 2123954 | Zbl 1202.47031
[13] F. Kittaneh: Notes on some inequalitis for Hilbert space operators. Publ. Res. Inst. Math. Sci. 24 (1988), 283-293. DOI 10.2977/prims/1195175202 | MR 0944864 | Zbl 0655.47009
[14] E. Nordgren, P. Rosenthal: Boundary values of Berezin symbols. Nonselfadjoint Operators and Related Topics (eds. A. Feintuch et al.), Oper. Theory, Adv. Appl. 73, Birkhäuser, Basel (1994), 362-368. DOI 10.1007/978-3-0348-8522-5_14 | MR 1320554 | Zbl 0874.47013
[15] A. Sheikhhosseini, M. S. Moslehian, K. Shebrawi: Inequalities for generalized Euclidean operator radius via Young's inequality. J. Math. Anal. Appl. 445 (2017), 1516-1529. DOI 10.1016/j.jmaa.2016.03.079 | MR 3545256 | Zbl 1358.47010
[16] K. Zhu: Operator Theory in Function Spaces. Pure and Applied Mathematics 139, Marcel Dekker, New York (1990). MR 1074007 | Zbl 0706.47019

Affiliations:   Mojtaba Bakherad, Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, University Blvd., P. O. Box 98155-987, Zahedan, Iran, e-mail: bakherad@member.ams.org

PDF available at: