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 on the classical Hardy space 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 in if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of is continuously extendable at one of the fixed points of and its value at the point is nonzero, then the cyclic subspace generated by is minimal if and only if is constant.

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1993-1152988-8

Article copyright:
© Copyright 1993
American Mathematical Society