Document Type

Article

Publication Date

2001

Publication Title

Linear Algebra and its Applications

Volume

331

Issue

1-3

First Page

61

Last Page

74

Abstract

Composition operators on the Hilbert Hardy space of the unit disk are considered. The shape of their numerical range is determined in the case when the symbol of the composition operator is a monomial or an inner function fixing 0. Several results on the numerical range of composition operators of arbitrary symbol are obtained. It is proved that 1 is an extreme boundary point if and only if 0 is a fixed point of the symbol. If 0 is not a fixed point of the symbol 1 is shown to be interior to the numerical range. Some composition operators whose symbol fixes 0 and has infinity norm less than 1 have closed numerical ranges in the shape of a cone-like figure, i.e. a closed convex region with a corner at 1, 0 in its interior, and no other corners. Compact composition operators induced by a univalent symbol whose fixed point is not 0 have numerical ranges without corners, except possibly a corner at 0.

Comments

Copyright © 2001 Elsevier Science Inc. All rights reserved.

This version of the article was released under a Creative Commons Attribution Non-Commercial No Derivatives License 3.0 (https://creativecommons.org/licenses/by-nc-nd/3.0/).

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