×

On some problems for operators on the reproducing kernel Hilbert space. (English) Zbl 07412210

Summary: In this study, some problems of operator theory on the reproducing kernel Hilbert space by using the Berezin symbols method are investigated. Namely, invariant subspaces of weighted composition operators on \(H^2\) are studied. Moreover, some new inequalities for the Berezin number of operators are proved. In particular, new reverse inequalities for the Berezin numbers \(\mathrm{ber} \big( |A|^2 \big)\) and \(\mathrm{ber} (A)\) of operators \(|A|^2 = A^{\ast} A\) and \(A\) on the reproducing kernel Hilbert space are given. Also, reverse inequalities for the Berezin number of two operators are proved. Under some conditions we prove the power inequality \[ \mathrm{ber} \big( A^n \big) \leq (\mathrm{ber} (A))^n \] which is related to a well known analogue estimate of Halmos for the numerical radius.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aronszajn, N., Theory of reproducing kernels, Trans Am Math Soc, 68, 3, 337-404 (1950) · Zbl 0037.20701
[2] Saitoh, S.Theory of reproducing kernels and its applications. England: Longman Scientific & Technical; 1988. p. 188. (Pitman research notes in mathematics series). · Zbl 0652.30003
[3] Saitoh, S.Integral transforms, reproducing kernels and their applications. Addison Wesley Longman; 1997. p. 369. (Pitman research notes in mathematics series). · Zbl 0891.44001
[4] Nikolski, NK.Operators, functions and systems: an easy reading. Volume I: Hardy, Hankel and Toeplitz. Dearborn: American Mathematical Society; 2002. (Mathematical surveys and monographs; vol. 92). · Zbl 1007.47001
[5] Nordgren, E.; Rosenthal, P., Boundary values of Berezin symbols, Oper Theory Adv Appl, 73, 362-368 (1994) · Zbl 0874.47013
[6] Zhu, K., Operator theory in function spaces, 258 (1990), New York: Marcel-Dekker Inc., New York · Zbl 0706.47019
[7] Karaev, MT., Reproducing kernels and Berezin symbols techniques in various questions of operator theory, Complex Anal Oper Theory, 7, 4, 983-1018 (2013) · Zbl 1303.47012
[8] Coburn, LA., A Lipschitz estimate for Berezin’s operator calculus, Proc Am Math Soc, 133, 127-131 (2004) · Zbl 1093.47024
[9] Axler, S.; Zheng, D., Compact operators via the Berezin transform, Indiana Univ Math J, 47, 387-400 (1998) · Zbl 0914.47029
[10] Bekolle, D.; Berger, CA; Coburn, LA, BMO in the Bergman metric on bounded symmetric domains, J Funct Anal, 93, 310-350 (1990) · Zbl 0765.32005
[11] Berger, CA; Coburn, LA., Toeplitz operators on the Segal-Bargmann space, Trans Am Math Soc, 301, 813-829 (1987) · Zbl 0625.47019
[12] Berger, CA; Coburn, LA., Heat flour and Berezin-Toeplitz estimates, Am J Math, 116, 563-590 (1994) · Zbl 0839.46018
[13] Engliš, M., Compact Toeplitz operators via the Berezin transform on bounded symmetric domains, Integral Equ Oper Theory, 33, 426-455 (1999) · Zbl 0936.47014
[14] Gullemin, V., Toeplitz operators in n dimensions, Integral Equ Oper Theory, 7, 145-205 (1984) · Zbl 0561.47025
[15] Garayev, MT; Guediri, H.; Sadraoui, H., Applications of reproducing kernels and Berezin symbols, N Y J Math, 22, 583-604 (2016) · Zbl 1350.47024
[16] Karaev, MT., Berezin symbol and invertibility of operators on the functional Hilbert spaces, J Funct Anal, 238, 181-192 (2006) · Zbl 1102.47018
[17] Cowen, C.; MacCluer, B., Composition operators on spaces of analytic functions (1995), Boca Raton: CRC Press, Boca Raton · Zbl 0873.47017
[18] Nordgren, E., Composition operators, Can J Math, 20, 442-449 (1968) · Zbl 0161.34703
[19] Shapiro, JH., What do composition operators know about inner functions?, Monatsh für Math, 130, 57-70 (2000) · Zbl 0951.47026
[20] Furuta, T.; Micic Hot, J.; Pečarić, J., Mond-Pečarić method in operator inequalities, Inequalities for bounded sef-adjoint operators on a Hilbert space (2005), Zagreb: Element, Zagreb · Zbl 1135.47012
[21] Dragomir, SS., A survey of some recent inequalities for the norm and numerical radius of operators in Hilbert spaces, Banach J Math Anal, 1, 154-175 (2007) · Zbl 1136.47006
[22] Hardy, GH., Divergent series (1973), London: Oxford University Press, London
[23] Yamancı, U.; Gürdal, M., On numerical radius and Berezin number inequalities for reproducing kernel Hilbert space, New York J Math, 23, 1531-1537 (2017) · Zbl 06865210
[24] Yamancı, U.; Garayev, MT; Çelik, C., Hardy-Hilbert type inequality in reproducing kernel Hilbert space: its applications and related results, Linear Multilinear Algebra, 67, 4, 830-842 (2019) · Zbl 07048425
[25] Kılıç, S., The Berezin symbol and multipliers of functional Hilbert spaces, Proc Am Math Soc, 123, 3687-3691 (1995) · Zbl 0847.46010
[26] Goldstein, A.; Ryff, JV; Clarke, LE., Problem 5473, Am Math Monthly, 75, 309 (1968)
[27] Dragomir, SS., A potpourri of Schwarz related inequalities in inner product spaces (II), J Ineq Pure Appl Math Art 14, 7, 1, 1-11 (2006) · Zbl 1137.46013
[28] Dragomir, SS; Sandor, J., Some inequalities in prehilbertian spaces, Studia Univ Babeş-Bolyai-Math, 32, 71-78 (1987) · Zbl 0634.46015
[29] Halmos, PR., A Hilbert space problem book (1982), New York: Springer-Verlag, New York · Zbl 0496.47001
[30] Garayev, MT; Gürdal, M.; Huban, MB., Reproducing kernels, Engliš algebras and some applications, Studia Math, 232, 2, 113-141 (2016) · Zbl 1372.47037
[31] Engliš, M., Toeplitz operators and the Berezin transform on \(####\), Linear Algebra Appl, 223-224, 171-204 (1995) · Zbl 0827.47017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.