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COMPACT DIFFERENCES OF COMPOSITION OPERATORS ON BERGMAN SPACES IN THE BALL

  • XIANG DONG YANG (a1) and LE HAI KHOI (a2)

Abstract

We obtain necessary and sufficient conditions for the compactness of differences of composition operators acting on the weighted Bergman spaces in the unit ball. A representation of a composition operator as a finite sum of composition operators modulo compact operators is also studied.

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Copyright

Corresponding author

For correspondence; e-mail: lhkhoi@ntu.edu.sg

Footnotes

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The first author is supported by the Natural Science Foundation of Yunnan Province in China (Grant No. 2009ZC013X) and the Basic Research Foundation of the Education Bureau of Yunnan Province in China (Grant No. 09Y0079).

Footnotes

References

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[1]Cowen, C. and MacCluer, B., Composition Operators on Spaces of Analytic Functions (CRC Press, Boca Raton, FL, 1995).
[2]Koo, H. and Smith, W., ‘Composition operators induced by smooth self-maps of the unit ball in ℂn’, J. Math. Anal. Appl. 329 (2007), 617633.
[3]Koo, H., Stessin, M. and Zhu, K., ‘Composition operators on the polydisc induced by smooth symbols’, J. Funct. Anal. 254 (2008), 29112925.
[4]Koo, H. and Wang, M., ‘Composition operators induced by smooth self-maps of the real or complex unit balls’, J. Funct. Anal. 256 (2009), 27472767.
[5]Kriete, T. L. and Moorhouse, J., ‘Linear relations in the Calkin algebra for composition operators’, Trans. Amer. Math. Soc. 359 (2007), 29152944.
[6]Moorhouse, J., ‘ C *-algebraic and component structure of composition operators’, PhD Thesis, University of Virginia, 2003.
[7]Moorhouse, J., ‘Compact differences of composition operators’, J. Funct. Anal. 219 (2005), 7092.
[8]Rudin, W., Function Theory in the Unit Ball of ℂn (Springer, New York, 1980).
[9]Shapiro, J. H., Compositions Operators and Classical Function Theory (Springer, New York, 1993).
[10]Shapiro, J. H. and Taylor, P. D., ‘Compact, nuclear and Hilbert Schmidt composition operators on H 2’, Indiana Univ. Math. J. 23 (1973), 471496.
[11]Wogen, W. R., ‘The smooth mappings which preserve the Hardy space H 2(𝔹n)’, Oper. Theory Adv. Appl. 35 (1988), 249263.
[12]Wogen, W. R., ‘On geometric properties of smooth maps which preserve H 2(𝔹n)’, Michigan Math. J. 54 (2006), 301306.
[13]Zhu, K., Operator Theory in Function Spaces, 2nd edn (American Mathematical Society, Providence, RI, 2007).
[14]Zhu, K., ‘Compact composition operators on Bergman spaces of the unit ball’, Houston J. Math. 33 (2007), 273283.
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COMPACT DIFFERENCES OF COMPOSITION OPERATORS ON BERGMAN SPACES IN THE BALL

  • XIANG DONG YANG (a1) and LE HAI KHOI (a2)

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