On the minimal invariant subspaces of the hyperbolic composition operator

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Abstract

The composition operator induced by a hyperbolic Möbius transform ϕ on the classical Hardy space H2 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 H2 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 ϕ and its value at the point is nonzero, then the cyclic subspace generated by u is minimal if and only if u is constant.

Original languageEnglish (US)
Pages (from-to)837-841
Number of pages5
JournalProceedings of the American Mathematical Society
Volume119
Issue number3
DOIs
StatePublished - Nov 1993

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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