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2009•Bulletin of the Australian Mathematical SocietyOpen access

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

Takuya Hosokawa

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Abstract

Abstract 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.

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Abstract 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.

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Abstract 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.

Key concepts: Mathematics, Bloch space, Infimum and supremum, Multiplication operator, Norm (philosophy), Operator norm, Composition operator, Operator (biology)

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