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COMPACT DIFFERENCES OF COMPOSITION OPERATORS

Published online by Cambridge University Press:  01 February 2008

ELKE WOLF*
Affiliation:
Mathematical Institute, University of Paderborn, D-33095 Paderborn, Germany (email: lichte@math.uni-paderborn.de)
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Abstract

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Let ϕ and ψ be analytic self-maps of the open unit disk. Each of them induces a composition operator, Cϕ and Cψ respectively, acting between weighted Bergman spaces of infinite order. We show that the difference CϕCψ is compact if and only if both operators are compact or both operators are not compact and the pseudohyperbolic distance of the functions ϕ and ψ tends to zero if ∣ϕ(z)∣→1 or ∣ψ(z)∣→1.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Bierstedt, K. D., Bonet, J. and Taskinen, J., ‘Associated weights and spaces of holomorphic functions’, Studia Math. 127 (1998), 137168.CrossRefGoogle Scholar
[2]Bonet, J., Domański, P., Lindström, M. and Taskinen, J., ‘Composition operators between weighted Banach spaces of analytic functions’, J. Austral. Math. Soc. Ser. A 64 (1998), 101118.CrossRefGoogle Scholar
[3]Bonet, J., Lindström, M. and Wolf, E., ‘Differences of composition operators between weighted Banach spaces of holomorphic functions’. J. Austral. Math. Soc. to appear.Google Scholar
[4]Cowen, C. and MacCluer, B., Composition operators on spaces of analytic functions (CRC Press, Boca Raton, FL, 1995).Google Scholar
[5]Lindström, M. and Wolf, E., ‘Essential norm of the difference of weighted composition operators’, Monatshefte Math. 153 (2008), 133143.CrossRefGoogle Scholar
[6]Lusky, W., ‘On weighted spaces of harmonic and holomorphic functions’, J. London Math. Soc. 51 (1995), 309320.CrossRefGoogle Scholar
[7]MacCluer, B., Ohno, S. and Zhao, R., ‘Topological structure of the space of composition operators on ’, Integral Equations Operator Theory 40(4) (2001), 481494.CrossRefGoogle Scholar
[8]Shapiro, J. H., Composition operators and classical function theory (Springer, Berlin, 1993).CrossRefGoogle Scholar