COMPOSITION OPERATORS ON ANALYTIC VECTOR-VALUED NEVANLINNA CLASSES

Abstract

Abstract:

Let ' be an analytic self-map of the complex unit disk and X a Banach space.
This paper studies the action of composition operator C' : f → f ? ' on the vector-valued
Nevanlinna classes N(X) and Na(X). Certain criteria for such operators to be weakly
compact are given. As a consequence, this paper shows that the composition operator
C' is weakly compact on N(X) and Na(X) if and only if it is weakly compact on the
vector-valued Hardy space H1(X) and Bergman space B1(X) respectively.

Key words: Composition operator, boundedness, weak compactness, Carleson measure,
vector-valued Nevanlinna class

CLC Number:

47B38 46E15

Wang Maofa. COMPOSITION OPERATORS ON ANALYTIC VECTOR-VALUED NEVANLINNA CLASSES[J].Acta mathematica scientia,Series B, 2005, 25(4): 771-780.

Trendmd

References

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

[2] Choa J S, Kim H O. Compact composition operators on the Nevanlinna class. Proc Amer Math Soc, 1997,
125: 145-151

[3] Choa J S, Kim H O, Shapiro J H. Compact composition operators on the Smirnov class. Proc Amer Math
Soc, 2000, 128: 2297-2308

[4] Choa J S, Kim H O. Compact composition operators on some F-algebras of holomorphic functions. Nihonkai
Math J, 1996, 7: 29-39

[5] Xiao J. Compact composition operators on the area-Nevanlinna class. Expositiones Math, 1999, 17:
255-264

[6] Choa J S, Kim H O. Composition operators between Nevanlinna-type spaces. J Math Anal Appl, 2001,
257: 378-402

[7] Shapiro J H. Composition Operators and Classical Function Theory. New York: Springer, 1993.

[8] Cowen C C, MacCluer B D. Composition Operators on Spaces of Analytic Functions. Boca Raton: CRC
Press, 1995.

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

[10] Bonet J, Doma´nski P, Lindstr¨om M. Weakly compact composition operators on analytic vector-valued
function spaces. Ann Acad Sci Fenn Math, 2001, 26 : 233-248

[11] Stoll M. Mean growth and Taylor coefficients of some topological algebra of analytic functions. Ann Polon
Math, 1977, 35: 139-158

[12] Shapiro J H, Shields A L. Unusual topological properties of the Nevanlinna class. Amer J Math, 1976, 97:
915-936

[13] Shapiro J H, Sundberg C. Compact composition operators on L1. Proc Amer Math Soc, 1990, 108:
443-449

[14] Stanton C S. Counting functions and majorization theorems for Jensen measures. Pacific J Math, 1986,
125: 459-468

[15] Katznelson Y. An Introduction to Harmonic Analysis. New York: Dover, 1976.

[16] Zhu K. Operator Theory in Function Spaces. New York: Marcel Dekker, 1990.

[17] Manuel D C, Alfredo G H. Weighted composition operators on Hardy spaces. J Math Anal Appl, 2001,
263: 224-233

[18] Aulaskari R, Direla D, Wulan H. Qp spaces, Hadamard products and Carleson measure. Math Reports.
2000, 2(52): 421-430

[19] MacCluer B D, Shapiro J H. Angular derivatives and compact composition operators on the Hardy and
Bergman spaces. Canadian J Math, 1986, 38: 878-906

[20] Wojtaszczyk P. Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics, Vol 25. Cambridge,U K: Cambrige Univ Press, 1991