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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Completely continuous composition operators

Authors: Joseph A. Cima and Alec Matheson
Journal: Trans. Amer. Math. Soc. 344 (1994), 849-856
MSC: Primary 47B38; Secondary 47B07
MathSciNet review: 1257642
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Abstract: A composition operator ${T_b}f = f \circ b$ is completely continuous on ${H^1}$ if and only if $|b| < 1$ a.e. If the adjoint operator $T_b^\ast$ is completely continuous on VMOA, then ${T_b}$ is completely continuous on ${H^1}$. Examples are given to show that the converse fails in general. Two results are given concerning the relationship between the complete continuity of an operator and of its adjoint in the presence of certain separability conditions on the underlying Banach space.

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Keywords: Composition operator, completely continuous operator
Article copyright: © Copyright 1994 American Mathematical Society