Bloch-type spaces and extended Cesàro operators in the unit ball of a complex Banach space. (English) Zbl 07069489
Summary: Let \(\mathbb{B}\) be the unit ball of a complex Banach space \(X\). In this paper, we generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball \(\mathbb{B}\) by using the radial derivative. Next, we define an extended Cesàro operator \(T_\varphi\) with holomorphic symbol \(\varphi\) and characterize those \(\varphi\) for which \(T_\varphi\) is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those \(\varphi\) for which \(T_\varphi\) is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol \(\varphi\). When \(\mathbb{B}\) is the open unit ball of a finite dimensional complex Banach space \(X\), this additional assumption is automatically satisfied.
MSC:
| 47B38 | Linear operators on function spaces (general) |
| 32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |
| 32A70 | Functional analysis techniques applied to functions of several complex variables |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
References:
| [1] | Aleman A, Cima J A. An integral operator on Hp and Hardy’s inequality. J Anal Math, 2001, 85: 157-176 · Zbl 1061.30025 · doi:10.1007/BF02788078 |
| [2] | Aleman A, Siskakis A G. Integration operators on Bergman spaces. Indiana Univ Math J, 1997, 46: 337-356 · Zbl 0951.47039 · doi:10.1512/iumj.1997.46.1373 |
| [3] | Blasco O, Galindo P, Lindström M, et al. Composition operators on the Bloch space of the unit ball of a Hilbert space. Banach J Math Anal, 2017, 11: 311-334 · Zbl 1361.32024 · doi:10.1215/17358787-0000005X |
| [4] | Blasco O, Galindo P, Miralles A. Bloch functions on the unit ball of an infinite dimensional Hilbert space. J Funct Anal, 2014, 267: 1188-1204 · Zbl 1293.32010 · doi:10.1016/j.jfa.2014.04.018 |
| [5] | Chu C-H, Hamada H, Honda T, et al. Bloch functions on bounded symmetric domains. J Funct Anal, 2017, 272: 2412-2441 · Zbl 1364.32004 · doi:10.1016/j.jfa.2016.11.005 |
| [6] | Deng F, Ouyang C. Bloch spaces on bounded symmetric domains in complex Banach spaces. Sci China Ser A, 2006, 49: 1625-1632 · Zbl 1114.32002 · doi:10.1007/s11425-006-2050-0 |
| [7] | Hamada H. Weighted composition operators from H1 to the Bloch space of infinite dimensional bounded symmetric domains. Complex Anal Oper Theory, 2018, 12: 207-216 · Zbl 06838043 · doi:10.1007/s11785-016-0624-6 |
| [8] | Hu Z. Extended Cesàro operators on the Bloch space in the unit ball of Cn. Acta Math Sci Ser B Engl Ed, 2003, 23: 561-566 · Zbl 1044.47023 · doi:10.1016/S0252-9602(17)30500-3 |
| [9] | Hu Z. Extended Cesàro operators on Bergman spaces. J Math Anal Appl, 2004, 296: 435-454 · Zbl 1072.47029 · doi:10.1016/j.jmaa.2004.01.045 |
| [10] | Hu Z, Wang S. Composition operators on Bloch-type spaces. Proc Roy Soc Edinburgh Sect A, 2005, 135: 1229-1239 · Zbl 1131.47019 · doi:10.1017/S0308210500004340 |
| [11] | Miao J. The Cesàro operator is bounded on Hp for 0 < p < 1. Proc Amer Math Soc, 1992, 116: 1077-1079 · Zbl 0787.47029 |
| [12] | Rudin W. Function theory in the unit ball of Cn. Grundlehren der Mathematischen Wissenschaften, vol. 241. New York-Berlin: Springer-Verlag, 1980 · Zbl 0495.32001 · doi:10.1007/978-1-4613-8098-6 |
| [13] | Shi J, Ren G. Boundedness of the Cesàro operator on mixed norm spaces. Proc Amer Math Soc, 1998, 126: 3553-3560 · Zbl 0905.47019 · doi:10.1090/S0002-9939-98-04514-6 |
| [14] | Siskakis A G. Composition semigroups and the Cesàro operator on Hp. J Lond Math Soc (2), 1987, 36: 153-164 · Zbl 0634.47038 · doi:10.1112/jlms/s2-36.1.153 |
| [15] | Siskakis A G. On the Bergman space norm of the Cesàro operator. Arch Math, 1996, 67: 312-318 · Zbl 0859.47024 · doi:10.1007/BF01197596 |
| [16] | Stević S. On an integral operator on the unit ball in Cn. J Inequal Appl, 2005, 2005: 81-88 · Zbl 1074.47013 |
| [17] | Tang X. Extended Cesàro operators between Bloch-type spaces in the unit ball of Cn. J Math Anal Appl, 2007, 326: 1199-1211 · Zbl 1117.47022 · doi:10.1016/j.jmaa.2006.03.082 |
| [18] | Wang S S, Hu Z J. Generalized Cesàro operators on Bloch-type spaces (in Chinese). Chinese Ann Math Ser A, 2005, 26: 613-624 · Zbl 1099.47507 |
| [19] | Wicker F D. Generalized Bloch mappings in complex Hilbert space. Canad J Math, 1977, 29: 299-306 · Zbl 0342.58010 · doi:10.4153/CJM-1977-033-9 |
| [20] | Xiao J. Cesàro-type operators on Hardy, BMOA and Bloch spaces. Arch Math, 1997, 68: 398-406 · Zbl 0870.30026 · doi:10.1007/s000130050072 |
| [21] | Xiao J. Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball. J Lond Math Soc (2), 2004, 70: 199-214 · Zbl 1064.47034 · doi:10.1112/S0024610704005484 |
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.
