Invariant subspaces of composition operators

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Abstract

The invariant subspace lattices of composition operators acting on H2, the Hilbert-Hardy space over the unit disc, are characterized in select cases. The lattice of all spaces left invariant by both a composition operator and the unilateral shift Mz (the multiplication operator induced by the coordinate function), is shown to be nontrivial and is completely described in particular cases. Given an analytic selfmap ϕ of the unit disc, we prove that ϕ has an angular derivative at some point on the unit circle if and only if Cϕ, the composition operator induced by ϕ, maps certain subspaces in the invariant subspace lattice of Mz into other such spaces. A similar characterization of the existence of angular derivatives of ϕ, this time in terms of Aϕ, the Aleksandrov operator induced by ϕ, is obtained.

Original languageEnglish (US)
Pages (from-to)243-264
Number of pages22
JournalJournal of Operator Theory
Volume73
Issue number1
DOIs
Publication statusPublished - Jan 1 2015

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Keywords

  • Composition operator
  • Invariant subspaces

ASJC Scopus subject areas

  • Algebra and Number Theory

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