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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
DOI: https://doi.org/10.1090/S0002-9939-1975-0377579-7
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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Article copyright: © Copyright 1975 American Mathematical Society