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COMPOSITION OPERATORS ON FINITE RANK MODEL SUBSPACES

Published online by Cambridge University Press:  02 August 2012

JAVAD MASHREGHI
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec G1K 7P4, Canada e-mail: [email protected]
MAHMOOD SHABANKHAH
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Québec H3A 2K6, Canada e-mail: [email protected]
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Abstract

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We give a complete description of bounded composition operators on model subspaces KB, where B is a finite Blaschke product. In particular, if B has at least one finite pole, we show that the collection of all bounded composition operators on KB has a group structure. Moreover, if B has at least two distinct finite poles, this group is finite and cyclic.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Beurling, A., On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239255.CrossRefGoogle Scholar
2.Chen, H. and Gauthier, P., Composition operators of μ-Bloch spaces, Canad. J. Math. 61 (2009), 5075.Google Scholar
3.Cowen, C. C., Composition operators on H 2, J. Operator Theory 9 (1983), 77106.Google Scholar
4.Cowen, C. C., Linear fractional composition operators on H 2, J. Integral Equ. Operator Theory 11 (1988), 151160.Google Scholar
5.Cowen, C. and MacCluer, B. D., Composition operators on spaces of analytic functions, in Studies in Advanced Mathematics, 1st edn. (CRC Press, Boca Raton, FL, 1995), 117221.Google Scholar
6.Frostman, O., Sur les produits de Blaschke (French), Kungl. Fysiografiska SŤ/llskapets i Lund Frhandlingar (Proc. Roy. Physiog. Soc. Lund) 12 (15) (1942), 169182. MR 0012127 (6:262e)Google Scholar
7.Ghatage, P., Zheng, D. and Zorboska, N., Sampling sets and closed range composition operators on the Bloch space, Proc. Amer. Math. Soc. 133 (2004), 13711377.Google Scholar
8.Gimémez, J., Malavé, R. and Ramos Fernàndez, J. C., Composition operators on μ-Bloch type spaces, Rend. Circ. Mat. Palermo 59 (2) (2010), 107119.Google Scholar
9.MacCluer, B., Compact composition operators in Hp(BN), Michigan Math. J. 32 (1985), 237248.Google Scholar
10.MacCluer, B. D. and Shapiro, J. H., Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. Math. J. 38 (1986), 878906.Google Scholar
11.MacCluer, B. D., Zeng, X. and Zorboska, N., Composition operators on small weighted Hardy spaces, Illinois J. Math. 40 (1996), 662667.Google Scholar
12.Needham, T., Visual complex analysis (Oxford University Press, Oxford, UK, 2002).Google Scholar
13.Shapiro, J. H., Composition operators and classical function theory, in Univeritext: Tracts in Mathematics (Springer-Verlag, 1993).Google Scholar
14.Shapiro, J. H. and Taylor, P. D., Compact, nuclear and Hilbert-Schmidt composition operators on H 2, Indiana Univ. Math. J. 23 (1973), 471496.Google Scholar
15.Tjani, M., Compact composition operators on Besov spaces, Trans. Amer. Math. Soc. 355 (11) (2003), 46834698.Google Scholar
16.Wulan, H., Zheng, D. and Zhu, K., Compact composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc. 137 (2009), 38613868.Google Scholar
17.Zorboska, N., Composition operators on weighted Dirichlet spaces, Proc. Amer. Math. Soc. 126 (7) (1998), 20132023.CrossRefGoogle Scholar