WEIGHTED COMPOSITION OPERATORS BETWEEN DIRICHLET SPACES

doi:10.1016/S0252-9602(11)60264-6

摘要/Abstract

摘要：

In this article, we study the boundedness of weighted composition operators between different vector-valued Dirichlet spaces. Some sufficient and necessary conditions for such operators to be bounded are obtained exactly, which are different completely from the scalar-valued case. As applications, we show that these vector-valued Dirichlet spaces are ifferent counterparts of the classical scalar-valued Dirichlet space and characterize the boundedness of multiplication operators between these different spaces.

关键词: Vector-valued analytic unction, Dirichlet space, weighted composition operator, boundedness

Abstract:

Key words: Vector-valued analytic unction, Dirichlet space, weighted composition operator, boundedness

中图分类号:

47B33

王茂发. WEIGHTED COMPOSITION OPERATORS BETWEEN DIRICHLET SPACES[J]. 数学物理学报(英文版), 2011, 31(2): 641-651.

WANG Mao-Fa. WEIGHTED COMPOSITION OPERATORS BETWEEN DIRICHLET SPACES[J]. Acta mathematica scientia,Series B, 2011, 31(2): 641-651.

参考文献

[1] Arregui J, Blasco O. Bergman and Bloch spaces of vector-valued functions. Math Nachr, 2003, 261/262: 3--22

[2] Barrenechea A, Pe~{n}a C. On Hadamard-Dirichlet algebras. Acta Math Univ Comenianae, 2002, 80:  9--17

[3] Blasco O. Boundary values of vector-valued harmonic functions considered as operators. Studia Math, 1987, 86: 19--33

[4] Blasco O. Remarks on vector-valued BMOA and vector-valued multipliers. Positivity, 2000, 4: 339--356

[5] Blasco O. Introduction to vector-valued Bergman spaces. University of Joensuu, Department of Mathematics, Report Series, 2005, 8: 9--30

[6] Bonet J, Doma\'{n}ski P, Lindstr\"{o}m M. Weakly compact composition operators on analytic vector-valued function spaces. Ann Acad Sci Fenn Math, 2001, 26: 233--248

[7] Cowen C, MacCluer B. Composition operators on spaces of analytic functions. Boca Raton: CRC Press, 1995

[8] Diestel J, Jarchow H, Tonge A. Absolutely summing operators. London: Cambridge Univ Press, 1995

[9] Dunford N, Schwartz J. Linear operators I. New York: John Wiley & Sons, 1958

[10]Halmos P. Measure Theory. New York: Springer-Verlag, 1974

[11] Hedenmalm H, Korenblum B, Zhu K. Theory of Bergman spaces. New York: Springer-Verlag, 2000

[12] Katznelson Y. An introduction to harmonic analysis. New York: Dover, 1976

[13]Kumar R,  Singh K. Essential normal of weighted composition operators on the Dirichlet space. Extracta Mathematicae, 2006, 21: 249--259

[14]Kumar S. Weighted composition operators between spaces of Dirichlet type. Rev Mat Complut, 2009, 22: 469--488

[15]Laitila J.   Weakly compact composition operators on vector-valued BMOA. J Math Anal Appl, 2005, 308: 730--745

[16] Laitila J, Tylli H -O. Composition operators on vector-valued harmonic functions and Cauchy transforms. Indiana Univ Math J, 2006,   55: 719--746

[17]Laitila J, Tylli H -O, Wang M. Composition operators from weak to strong spaces of vector-valued analytic functions. J Operator Theory, 2009, 62: 281--295

[18] Luecking D. Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives. Amer J Math, 1985, 107: 85--111

[19]Liu P, Saksman E, Tylli H -O. Small composition operators on analytic vector-valued function spaces. Pacific J Math, 1998, 184: 295--309

[20]MacCluer B, Shapiro J. Angular derivatives and compact composition operators on the Hardy and Dergman spaces. Canad J Math, 1986, 38: 878--906

[21] Mirzakarimi G, Seddighi K. Weighted composition operators on Bergman and Dirichlet spaces. Georgian Math J, 1997, 4: 373--383

[22]Shapiro J. The essential norm of a composition operator. Annals Math, 1987, 125: 375--404

[23]Shapiro J. Composition operators and classical function theory. New York: Springer-verlag, 1993

[24] Smith W. Composition operators between Bergman and Hardy spaces. Trans Amer Math Soc, 1996, 248: 2331--2348

[25] Tjani M. Compact composition operators on some M\"{o}bius invariant Banach spaces. Ph D.-thesis. Michigan State University, 1996

[26] Vukotic D. On the coefficient multipliers of Bergman spaces. J London Math Soc, 1994, 50: 341--348

[27]Wu Z. Carleson measures and multipliers for Dirichlet spaces. J Funct Anal, 1999, 169: 148--163

[28]Zhou Z, Yuan C. Linear fractional composition operators on the Dirichlet space in the unit ball. Sci China, 2009, 52A: 1661--1670

[29] Zorboska N. Composition operators on weighted Dirichlet spaces. Proc Amer Math Soc, 1998, 126: 2013--2023