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Multiplicative Isometries and Isometric Zero-Divisors

Published online by Cambridge University Press:  20 November 2018

Alexandru Aleman ,
Peter Duren ,
María J. Martín  and
Dragan Vukotić
Alexandru Aleman*
Affiliation:
Department of Mathematics, Lund University, P. O. Box 118, S-221 00 Lund, Sweden
Peter Duren*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1043, USA
María J. Martín*
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Dragan Vukotić*
Affiliation:
Departamento de Matemáticas & ICMAT CSIC-UAM-UC3M-UCM, Universidad Autónoma de Madrid, 28049 Madrid, Spain
*
aleman@maths.lth.se
duren@umich.edu
mariaj.martin@uam.es
dragan.vukotic@uam.es
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Abstract

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For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.

Keywords

30H0546E15Banach spaces of analytic functionsHardy spacesBergman spacesBloch spaceDirichlet spaceDirichlet-type spacespointwise multiplierscoefficient multipliersisometriesisometric zerodivisors
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 5 , 01 October 2010 , pp. 961 - 974
Copyright
Copyright © Canadian Mathematical Society 2010

Footnotes

The research of the second, third, and fourth authors is supported by MICINN grant MTM2009-14694-C02-01, Spain. The third and fourth authors are partially supported by the Thematic Network MTM2008-02829-E fromMICINN. The fourth author is also partially supported by the European Science Foundation Network HCAA (“Harmonic and Complex Analysis and Applications”).

References

[1] Brown, L. and Shields, A. L., Multipliers and cyclic vectors in the Bloch space. Michigan Math. J. 38(1991), 141–146. doi:10.1307/mmj/1029004269Google Scholar
[2] Buckley, S. M., P. Koskela and D. Vukotić, Fractional integration, differentiation, and weighted Bergman spaces. Math. Proc. Cambridge Philos. Soc. 126(1999), 369–385. doi:10.1017/S030500419800334XGoogle Scholar
[3] Cima and W, J. A..Wogen, On isometries of the Bloch space. Illinois J. Math. 24(1980), 313–316.Google Scholar
[4] Colonna, F., Characterisation of the isometric composition operators on the Bloch space. Bull. Austral. Math. Soc. 72(2005), 283–290. doi:10.1017/S0004972700035073Google Scholar
[5] Duren, P. L., Theory of Hp Spaces. Academic Press, New York, 1970; reprinted with supplement by Dover Publications, Mineola, NY, 2000.Google Scholar
[6] Duren, P. and Schuster, A., Bergman Spaces. American Mathematical Society, Providence, RI, 2004.Google Scholar
[7] Duren, P., Khavinson, D., Shapiro, H. S. and Sundberg, C., Contractive zero-divisors in Bergman spaces. Pacific J. Math. 157(1993), 37–56.Google Scholar
[8] Forelli, F., The isometries of Hp. Canad. J. Math. 16(1964), 721–728.Google Scholar
[9] Hedenmalm, H., A factorization theorem for square area-integrable analytic functions. J. Reine Angew. Math. 422(1991), 45–68.Google Scholar
[10] Kolaski, C. J., Isometries of weighted Bergman spaces. Canad. J. Math. 34(1982), 910–915.Google Scholar
[11] Martın, M. J. and Vukotić, D., Isometries of the Dirichlet space among the composition operators. Proc. Amer. Math. Soc. 134(2006), 1701–1705. doi:10.1090/S0002-9939-05-08182-7Google Scholar
[12] Martın, M. J. and Vukotić, D., Isometries of some classical function spaces among the composition operators. Contemp. Math. 393(2006), 133–138.Google Scholar
[13] Martın, M. J. and Vukotić, D., Isometries of the Bloch space among the composition operators. Bull. London Math. Soc. 39(2007), 151–155.Google Scholar
[14] Stegenga, D., Multipliers of the Dirichlet space. Illinois J. Math. 24(1980), 113–139.Google Scholar