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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Spectra of compact composition operators

Authors: James G. Caughran and Howard J. Schwartz
Journal: Proc. Amer. Math. Soc. 51 (1975), 127-130
MSC: Primary 47B37; Secondary 30A18
MathSciNet review: 0377579
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Abstract: Let $ \phi $ be holomorphic and map the open unit disk into itself, and let $ {C_\phi }:f \to f \circ \phi $ be the composition operator on $ {H^2}$ generated by $ \phi $. If $ {C_\phi }$ is a compact operator then $ (1)\phi ({z_0}) = {z_0}$ for some $ {z_0} \epsilon D$; $ (2)\sigma ({C_\phi }) = \{ \phi '{({z_0})^n}:n = 0,1,2, \ldots \} \cup \{ 0\} $.

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PII: S 0002-9939(1975)0377579-7
Article copyright: © Copyright 1975 American Mathematical Society