Refinements of Kantorovich type, Schwarz and Berezin number inequalities
DOI:
https://doi.org/10.17398/2605-5686.35.1.1Keywords:
Reproducing kernel Hilbert space, Berezin symbol, Berezin number, Kantorovich type inequality, C∗ -moduleAbstract
In this article, we use Kantorovich and Kantorovich type inequalities in order to prove some new Berezin number inequalities. Also, by using a refinement of the classical Schwarz inequality, we prove Berezin number inequalities for powers of f (A), where A is self-adjoint operator on the Hardy space H 2(D) and f is a positive continuous function. Some related questions are also discussed.
Downloads
References
T. Ando, C.-K. Li, R. Mathias, Geometric means, Linear Algebra Appl. 385 (2004), 305 – 334.
L. Arambas̆ić, D. Bakić, M.S. Moslehian, A treatment of the Cauchy-Schwarz inequality in C ∗ -Modules, J. Math. Anal. Appl. 381 (2011), 546 – 556.
N. Aronzajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337 – 404.
M. Bakherad, Some Berezin number inequalities for operator matrices, Czechoslovak Math. J., 68 (4) (2018), 997 – 1009.
M. Bakherad, M. Garayev, Berezin number inequalities for operators, Concr. Oper. 6 (1) (2019), 33 – 43.
H. Başaran, M. Gürdal, A.N. Güncan, Some operator inequalities associated with Kantorovich and Hölder-McCarthy inequalities and their applications, Turkish J. Math. 43 (1) (2019), 523 – 532.
F.A. Berezin, Covariant and contravariant symbols for operators, Math. USSR-Izv. 6 (1972), 1117 – 1151.
F.A. Berezin, Quantization, Math. USSR-Izv. 8 (1974), 1109 – 1163.
L.A. Coburn, Berezin transform and Weyl-type unitary operators on the Bergman space, Proc. Amer. Math. Soc. 140 (2012), 3445 – 3451.
S.S. Dragomir, New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces, Linear Algebra Appl. 428 (11-12) (2008), 2750 – 2760.
S.S. Dragomir, Improving Schwartz inequality in inner product spaces, Linear Multilinear Algebra 67 (2) (2019), 337 – 347.
M. Enomoto, Commutative relations and related topics, Kyoto University, 1039 (1998), 135 – 140.
T. Furuta, Operator inequalities associated with Hölder–McCarthy and Kantorovich inequalities, J. Inequal. Appl. 2 (2) (1998), 137 – 148.
T. Furuta, “ Invitation to Linear Operators. From matrices to bounded linear operators on a Hilbert space ”, Taylor & Francis, London, 2001 (p. 266).
T. Furuta, J. Mićić Hot, J.E. Pečarić, Y. Seo, Mond-Pecaric method in operator inequalities, in “ Inequalities for bounded selfadjoint operators on a Hilbert space ”, Monographs in Inequalities, Vol. 1, Element, Zagreb, 2005.
M.T. Garayev, M. Gürdal, M.B. Huban, Reproducing kernels, Engliš algebras and some applications, Studia Math. 232 (2) (2016), 113 – 141.
M.T. Garayev, M. Gürdal, A. Okudan, Hardy-Hilbert’s inequality and power inequalities for Berezin numbers of operators, Math. Inequal. Appl. 19 (3) (2016), 883 – 891.
M.T. Garayev, M. Gürdal, S. Saltan, Hardy type inequality for reproducing kernel Hilbert space operators and related problems, Positivity 21 (4) (2017), 1615 – 1623.
M.T. Garayev, Berezin symbols, Hölder-McCarthy and Young inequalities and their applications, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 43 (2) (2017), 287 – ,295.
W. Greub, W. Rheinboldt, On a generalization of an inequality of L.V. Kantorovich, Proc. Amer. Math. Soc. 10 (1959), 407 – 415.
M. Gürdal, M.T. Garayev, S. Saltan, U. Yamancı, On some numerical characteristics of operators, Arab J. Math. Sci. 21 (1) (2015), 118 – 126.
P. Halmos, “ A Hilbert Space Problem Book ”, Graduate Texts in Mathematics, Vol. 19, Springer-Verlag, New York-Berlin, 1982.
K. Hoffman, “ Banach Spaces of Analytic Functions ”, Dover Publications, Mineola, New York, 2007.
L.V. Kantorović, Functional analysis and applied mathematics, Uspehi Matem. Nauk (N.S.) 3 (6(28)) (1948), 89 – 185 (in Russian).
M.T. Karaev, Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal. Oper. Theory 7 (4) (2013), 983 – 1018.
M.T. Karaev, S. Saltan, Some results on Berezin symbols, Complex Var. Theory Appl. 50 (2005), 185 – 193.
M. Kian, Hardy-Hilbert type inequalities for Hilbert space operators, Ann. Funct. Anal. 3 (2) (2012), 129 – 135.
S. Kiliç, The Berezin symbol and multipliers of functional Hilbert spaces, Proc. Amer. Math. Soc. 123 (1995), 3687 – 3691.
E.C. Lance, “ Hilbert C ∗-modules ”, London Math. Soc. Lecture Notes Series, 210, Cambridge University Press, Cambridge, 1995.
S. Liu, H. Neudecker, Several matrix Kantorovich-type inequalities, J. Math. Anal. Appl. 197 (1996), 23 – 26.
V.M. Manuilov, E.V. Troitsky, “ Hilbert C ∗-modules ”, Translations of Mathematical Monographs, 226, American Mathematical Society, Providence RI, , 2005.
A.W. Marshall, I. Olkin, Matrix versions of the Cauchy and Kantorovich inequalities, Aequationes Math. 40 (1) (1990), 89 – 93.
D.S. Mitrinovic, J.E. Pečarić, A.M. Fink, “Classical and new inequalities in analysis”, Kluwer Academic Publishers Group, Dordrecht, 1993.
M.S. Moslehian, Recent developments of the operator Kantorovich inequality, Expo. Math. 30 (2012), 376 – 388.
R. Nakamoto, M. Nakamura, Operator mean and Kantorovich inequality, Math. Japon. 44 (3) (1996), 495 – 498.
M. Niezgoda, Kantorovich type inequalities for ordered linear spaces, Electron. J. Linear Algebra 20 (2010), 103 – 114.
C. Pearcy, An elementary proof of the power inequality for the numerical radius, Michigan Math. J. 13 (1966), 289 – 291.
G. Strang, On the Kantorovich inequality, Proc. Amer. Math. Soc. 11 (1960), 468.
U. Yamancı, M.T. Garayev, C. Çelik, Hardy-Hilbert type inequality in reproducing kernel Hilbert space: its applications and related results, Linear Multilinear Algebra 67 (4) (2019), 830 – 842.
U. Yamancı, M. Gürdal, M.T. Garayev, Berezin number inequality for convex function in reproducing kernel Hilbert space, Filomat 31 (18) (2017), 5711 – 5717.
T. Yamazaki, An extension of Kantorovich inequality to n-operators via the geometric mean by Ando-Li-Mathias, Linear Algebra Appl. 416 (2-3) (2006), 688 – ,695.
T. Yamazaki, M. Yanagida, Characterizations of chaotic order associated with Kantorovich inequality, Sci. Math. 2 (1) (1999), 37 – 50.
H. Umegaki, Conditional expectation in an operator algebra, Tohoku Math. J. (2) 6 (1954), 177 – 181.
F. Zhang, Equivalence of the Wielandt inequality and the Kantorovich inequality, Linear Multilinear Algebra 48 (3) (2001), 275 – 279.
