Nonminimal cyclic invariant subspaces of hyperbolic composition operators

Research output: Chapter in Book/Report/Conference proceedingChapter

1 Scopus citations

Abstract

Operators on function spaces acting by composition to the right with a fixed self-map ϕ are called composition operators. We denote them Cϕ. Given ϕ, a hyperbolic disc automorphism, the composition operator Cϕ on the Hilbert Hardy space H2 is considered. The bilateral cyclic invariant subspaces Kf, f ∈ H2, of Cϕ are studied, given their connection with the invariant subspace problem, which is still open for Hilbert space operators. We prove that nonconstant inner functions u induce non–minimal cyclic subspaces Ku if they have unimodular, orbital, cluster points. Other results about Ku when u is inner are obtained. If f ∈ H2 \ {0} has a bilateral orbit under Cϕ, with Cesàro means satisfying certain boundedness conditions, we prove Kf is non–minimal invariant under Cϕ. Other results proving the non–minimality of invariant subspaces of Cϕ of type Kf when f is not an inner function are obtained as well.

Original languageEnglish (US)
Title of host publicationContemporary Mathematics
PublisherAmerican Mathematical Society
Pages247-262
Number of pages16
DOIs
StatePublished - 2017

Publication series

NameContemporary Mathematics
Volume699
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627

Keywords

  • Composition operators
  • Invariant subspaces

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Matache, V. (2017). Nonminimal cyclic invariant subspaces of hyperbolic composition operators. In Contemporary Mathematics (pp. 247-262). (Contemporary Mathematics; Vol. 699). American Mathematical Society. https://doi.org/10.1090/conm/699/14094