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THE ESSENTIAL NORM OF A WEIGHTED COMPOSITION OPERATOR FROM THE BLOCH SPACE TO H∞

Part of: Special classes of linear operators

Published online by Cambridge University Press:  05 May 2009

TAKUYA HOSOKAWA
TAKUYA HOSOKAWA*
Affiliation:
Institute of Basic Science, Korea University, Seoul, 136-713, Korea (email: [email protected])
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Abstract

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We express the operator norm of a weighted composition operator which acts from the Bloch space ℬ to H∞ as the supremum of a quantity involving the weight function, the inducing self-map, and the hyperbolic distance. We also express the essential norm of a weighted composition operator from ℬ to H∞ as the asymptotic upper bound of the same quantity. Moreover we study the estimate of the essential norm of a weighted composition operator from H∞ to itself.

Keywords

weighted composition operatorthe space of bounded analytic functionsBloch space

MSC classification

Secondary: 47B33: Composition operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 3 , June 2009 , pp. 487 - 497
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This work was supported by the Korean Research Foundation Grant funded by Korean Government (KRF-2008-314-C00012).

References

[1] Cowen, C. C. and MacCluer, B. D., Composition Operators on Spaces of Analytic Functions (CRC Press, Boca Raton, 1995)Google Scholar
[2] deLeeuw, K. and Rudin, W., ‘Extreme points and extreme problems in H 1’, Pacific J. Math. 8 (1958), 467–485CrossRefGoogle Scholar
[3] Hoffman, K., Banach Spaces of Analytic Functions (Prentice Hall, Englewood Cliffs, NJ, 1962)Google Scholar
[4] Hosokawa, T., Izuchi, K. and Ohno, S., ‘Topological structure of the space of weighted composition operators on H ∞’, Integral Equations Operator Theory 53 (2005), 509–526CrossRefGoogle Scholar
[5] Kwon, E. G., ‘Hyperbolic g-function and Bloch pullback operators’, J. Math. Anal. Appl. 309 (2005), 626–637CrossRefGoogle Scholar
[6] Ohno, S., ‘Weighted composition operators between H ∞ and the Bloch space’, Taiwanese J. Math. 5 (2001), 555–563CrossRefGoogle Scholar
[7] Takagi, H., Takahashi, J. and Ueki, S., ‘The essential norm of a weighted composition operator on the ball algebra’, Acta Sci. Math. (Szeged) 70 (2004), 819–829Google Scholar
[8] Zheng, L., ‘The essential norms and spectra of composition operators on H ∞’, Pacific J. Math. 203 (2002), 503–510CrossRefGoogle Scholar
[9] Zhu, K., Operator Theory in Function Spaces (Marcel Dekker, New York, 1990)Google Scholar