Articles

WEIGHTED COMPOSITION OPERATORS BETWEEN DIRICHLET SPACES

  • WANG Mao-Fa
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  • School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Online published: 2011-03-20

Supported by

This work is partially supported by the National Natural Science Foundation of China (10901158)

Abstract

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.

Cite this article

WANG Mao-Fa . WEIGHTED COMPOSITION OPERATORS BETWEEN DIRICHLET SPACES[J]. Acta mathematica scientia, Series B, 2011 , 31(2) : 641 -651 . DOI: 10.1016/S0252-9602(11)60264-6

References

[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 

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