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ISOMETRIES AND SPECTRA OF MULTIPLICATION OPERATORS ON THE BLOCH SPACE

Part of: Special classes of linear operators

Published online by Cambridge University Press:  09 February 2009

ROBERT F. ALLEN  and
FLAVIA COLONNA
ROBERT F. ALLEN
Affiliation:
Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA (email: rallen2@gmu.edu)
FLAVIA COLONNA*
Affiliation:
Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA (email: fcolonna@gmu.edu)
*
✉For correspondence; e-mail: fcolonna@gmu.edu
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Abstract

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In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identify the isometries and spectra of a particular class of weighted composition operators on the Bloch space.

Keywords

multiplication operatorsBloch spacelittle Bloch space

MSC classification

Secondary: 47B38: Operators on function spaces (general)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 1 , February 2009 , pp. 147 - 160
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Anderson, J. M., Clunie, J. and Pommerenke, Ch., ‘On Bloch functions and normal functions’, J. Reine Angew. Math. 279 (1974), 12–37.Google Scholar
[2] Brown, L. and Shields, A. L., ‘Multipliers and cyclic vectors in the Bloch space’, Michigan Math. J. 38 (1991), 141–146.CrossRefGoogle Scholar
[3] Cima, J., ‘The basic properties of Bloch functions’, Int. J. Math. Math. Sci. 3 (1979), 369–413.CrossRefGoogle Scholar
[4] Cima, J. and Wogen, W., ‘On isometries of the Bloch space’, Illinois J. Math. 24 (1980), 313–316.CrossRefGoogle Scholar
[5] Colonna, F., ‘Extreme points of a convex set of Bloch functions’, Seminars in Complex Analysis and Geometry, Sem. Conf. 4 (1988), 23–59.Google Scholar
[6] Colonna, F., ‘Characterisation of the isometric composition operators on the Bloch space’, Bull. Aust. Math. Soc. 72 (2005), 283–290.CrossRefGoogle Scholar
[7] Conway, J. B., A Course in Functional Analysis, 2nd edn (Springer, New York, 1990).Google Scholar
[8] Douglas, R. G., Banach Algebra Techniques in Operator Theory, 2nd edn (Springer, New York, 1998).CrossRefGoogle Scholar
[9] Duren, P. L., Romberg, B. W. and Shields, A. L., ‘Linear functionals on H p spaces with 0<p<1’, J. Reine Angew. Math. 238 (1969), 32–60.Google Scholar
[10] El-Gebeily, M. and Wolfe, J., ‘Isometries of the disc algebra’, Proc. Amer. Math. Soc. 93 (1985), 697–702.Google Scholar
[11] Forelli, F., ‘The isometries on H p’, Canad. J. Math. 16 (1964), 721–728.CrossRefGoogle Scholar
[12] Gorkin, P. and Mortini, R., ‘Universal Blaschke products’, Proc. Cambridge Philos. Soc. 136 (2004), 175–184.CrossRefGoogle Scholar
[13] Hedenmalm, H., ‘Thin interpolating sequences and three algebras of bounded functions’, Proc. Amer. Math. Soc. 99 (1987), 489–495.CrossRefGoogle Scholar
[14] Kolaski, C. J., ‘Isometries of weighted Bergman spaces’, Canad. J. Math 34 (1982), 910–915.CrossRefGoogle Scholar
[15] Martín, M. J. and Vukotić, D., ‘Isometries of the Bloch space among the composition operators’, Bull. London Math. Soc. 39 (2007), 151–155.Google Scholar
[16] Ohno, S. and Zhao, R., ‘Weighted composition operators on the Bloch space’, Bull. Aust. Math. Soc. 63 (2001), 177–185.CrossRefGoogle Scholar
[17] Pommerenke, Ch., ‘On Bloch functions’, J. London Math. Soc. 2 (1970), 689–695.CrossRefGoogle Scholar
[18] Rudin, W., ‘L p-isometries and equimeasurability’, Indiana Univ. Math. J. 25 (1976), 215–228.CrossRefGoogle Scholar
[19] Xiong, C., ‘Norm of composition operators on the Bloch space’, Bull. Aust. Math. Soc. 70 (2004), 293–299.Google Scholar