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

On the minimal invariant subspaces of the hyperbolic composition operator


Author: Valentin Matache
Journal: Proc. Amer. Math. Soc. 119 (1993), 837-841
MSC: Primary 47B38; Secondary 46E20, 47A15
MathSciNet review: 1152988
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Abstract: The composition operator induced by a hyperbolic Möbius transform $ \phi $ on the classical Hardy space $ {H^2}$ is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function $ u$ in $ {H^2}$ if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of $ u$ is continuously extendable at one of the fixed points of $ \phi $ and its value at the point is nonzero, then the cyclic subspace generated by $ u$ is minimal if and only if $ u$ is constant.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1152988-8
PII: S 0002-9939(1993)1152988-8
Article copyright: © Copyright 1993 American Mathematical Society