Abstract
The Berezin transform à of an operator A, acting on the reproducing kernel Hilbert space ℋ = ℋ (Ω) over some (non-empty) set Ω, is defined by Ã(λ) = 〉Aǩ λ, ǩ λ〈 (λ ∈ Ω), where
and
We also prove the multiplicative inequality
References
[1] Abu-Omar A. and Kittaneh F., Numerical radius inequalities for n × n operator matrices, Linear Algebra Appl. 468 (2015), 18–2610.1016/j.laa.2013.09.049Search in Google Scholar
[2] Abu-Omar A. and Kittaneh F., Numerical radius inequalities for products and commutators of operators, Houston J. Math. 41 (2015), no. 4, 1163–1173Search in Google Scholar
[3] Ando T. and Zhan X., Norm inequalities related to operator monotone functions, Math. Ann. 315 (1999), no. 4, 771–780.Search in Google Scholar
[4] Berezin F.A., Covariant and contravariant symbols for operators, Math. USSR-Izv. 6 (1972) 1117–1151.10.1070/IM1972v006n05ABEH001913Search in Google Scholar
[5] Berezin F.A., Quantization, Math. USSR-Izv. 8 (1974) 1109–1163.10.1070/IM1974v008n05ABEH002140Search in Google Scholar
[6] Engli M.š, Toeplitz operators and the Berezin transform on H2, Special issue honoring Miroslav Fiedler and Vlastimil Pták. Linear Algebra Appl. 223/224 (1995), 171–204.10.1016/0024-3795(94)00056-JSearch in Google Scholar
[7] Garayev M.T., Gürdal M. and Okudan A., Hardy-Hilbert’s inequality and power inequalities for Berezin numbers of operators, Math. Inequal. App., 19 (2016), no. 3, 883–891Search in Google Scholar
[8] Garayev M.T., Gürdal M., and Saltan S., Hardy type inequality for reproducing kernel Hilbert space operators and related problems, Positivity, 21 (2017), no. 4, 1615–1623.Search in Google Scholar
[9] Gustafson K.E. and Rao D.K.M., Numerical Range, The Field of Values of Linear Operators andMatrices, Springer, New York, 1997.10.1007/978-1-4613-8498-4_1Search in Google Scholar
[10] Hajmohamadi M., Lashkaripour R. and Bakherad M., Improvements of Berezin number inequalities, Linear and Multilinear Algebra, (to appear), doi.org/10.1080/03081087.2018.1538310Search in Google Scholar
[11] Halmos P.R., A Hilbert space problem book, Springer Verlag, New York, 1982.10.1007/978-1-4684-9330-6Search in Google Scholar
[12] Karaev M.T., Berezin symbol and invertibility of operators on the functional Hilbert spaces, J. Funct. Anal., 238 (2006) 181–192.Search in Google Scholar
[13] Karaev M.T., On the Berezin symbol, J. Math. Sci. (New York) 115 (2003) 2135–2140. Translated from: Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 270 (2000) 80–89.Search in Google Scholar
[14] Karaev M.T., Functional analysis proofs of Abel’s theorems, Proc. Amer. Math. Soc. 132 (2004) 2327–2329.10.1090/S0002-9939-04-07354-XSearch in Google Scholar
[15] Karaev M.T. and Saltan S., Some results on Berezin symbols, Complex Var. Theory Appl. 50 (3) (2005) 185–193.10.1080/02781070500032861Search in Google Scholar
[16] Karaev M.T., Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory 7 (2013), no. 4, 983–1018.Search in Google Scholar
[17] Karaev M.T., Saltan S. and Gundogdu D., On the inverse power inequality for the Berezin number of operators, J. Math. Inequal. 12 (2018), no. 4, 997–1003.Search in Google Scholar
[18] Kili S., c, The Berezin symbol and multipliers of functional Hilbert spaces, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3687–3691.Search in Google Scholar
[19] Kittaneh F., Notes on some inequalitis for Hilbert space operators, Publ. Res. Inst. Math. Sci. 24 (2) (1988), 283–293.10.2977/prims/1195175202Search in Google Scholar
[20] Nordgren E. and Rosenthal P., Boundary values of Berezin symbols, Oper. Theory Adv. Appl. 73 (1994) 362–368.Search in Google Scholar
[21] Yamanci U. and Gürdal M., On numerical radius and Berezin number inequalities for reproducing kernel Hilbert space, New York J. Math. 23 (2017), 1531–1537.Search in Google Scholar
[22] Yamanci U., Gürdal M. and Garayev M.T., Berezin number inequality for convex function in reproducing Kernel Hilbert space, Filomat 31 (2017), no. 18, 5711–5717.Search in Google Scholar
[23] Yamazaki T., On upper and lower bounds of the numerical radius and an equality condition, StudiaMath. 178 (2007), 83–89.Search in Google Scholar
[24] Zhu K., Operator Theory in Function Spaces, Dekker, New York, 1990.Search in Google Scholar
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