Acta mathematica scientia, Series B >
COMPACT DIFFERENCES OF COMPOSITION OPERATORS ON HOLOMORPHIC FUNCTION SPACES IN THE UNIT BALL
Received date: 2010-08-16
Online published: 2011-09-20
We find a lower bound for the essential norm of the difference of two compo-sition operators acting on H2(BN) or A2s(BN) (s > −1). This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.
Key words： composition operators; Hardy space; Bergman spaces; compact differences
JIANG Liang-Ying , OUYANG Cai-Heng . COMPACT DIFFERENCES OF COMPOSITION OPERATORS ON HOLOMORPHIC FUNCTION SPACES IN THE UNIT BALL[J]. Acta mathematica scientia, Series B, 2011 , 31(5) : 1679 -1693 . DOI: 10.1016/S0252-9602(11)60353-6
[1] Jiang L Y. Linear Fractional Composition Operators on Holomorphic Function Spaces [D]. Chinese Academy of Sciences, 2008 (in Chinese)
[2] Rudin W. Function Theory in the Unit Ball of CN. New York: Springer-Verlag, 1980
[3] Zhu K H. Spaces of Holomorphic Functions in the Unit Ball. New York: Springer-Verlag, 2005
[4] Shapiro J H. Composition Operators and Classical Function Theory. New York: Springer-Verlag, 1993
[5] Cowen C C, MacCluer B D. Composition Operators on Spaces of Analytic Functions. Boca Raton: CRC Press, 1995
[6] Berkson E. Composition operators isolated in the uniform operator topology. Proc Amer Math Soc, 1981, 81: 230–232
[7] Shapiro J H, Sundberg C. Isolation amongst the composition operators. Pacific J Math, 1990, 145: 117–152
[8] Heidler H. Algebraic and essentially algebraic composition operators on the ball or polydisk//Studies on Composition Operators (Laramie, 1996). Contemporary Math, 213. Amer Math Soc, 1998: 43–56
[9] Hammond C, MacCluer B D. Isolation and component structure in spaces of compoaition operators. Integr Equ Oper Theory, 2005, 53: 269–285
[10] MacCluer B D. Components in the space of composition operators. Integr Equ Oper Theory, 1989, 12: 725–738
[11] Gallardo-Guti´errez E A, Gonz´alez M J, Nieminen P J, Saksman E. On the connected component of compact composition operators on the Hardy space. Adv Math, 2008, 219: 986–1001
[12] Kriete T, Moorhouse J. Linear relations in the Calkin algebra for composition operators. Trans Amer Math Soc, 2007, 359: 2915-2944
[13] Shapiro J H. Aleksandrov measures used in essential norm inequalities for composition operators. J Oper Theory, 1998, 40: 133–146
[14] Bourdon P S. Components of linear fractional composition operators. J Math Anal Appl, 2003, 279: 228-245
[15] Moorhouse J. Compact differences of composition operators. J Funct Anal, 2005, 219: 70-92
[16] MacCluer B D, Weir R J. Linear-fractional composition operators in several variables. Integr Equ Oper Theory, 2005, 53: 373–402
[17] Jiang L Y, Ouyang C H. Essential normality of linear fractional composition operators in the unit ball of CN. Sci China Series A, 2009, 52: 2668–2678
[18] Cowen C C, MacCluer B D. Linear fractional maps of the unit ball and their composition operators. Acta Sci Math (Szeged), 2000, 66: 351–376
/
| 〈 |
|
〉 |