Maximal, minimal, and primary invariant subspaces

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Let X be a complex infinite dimensional Banach space. An operator L on X is called of subcritical class, if ∑n=1 n-3/2 log +∥Ln∥ < ∞. Assume that T is an operator on X whose iterates have norms of polynomial growth. We prove that if T has a range of finite codimension and a left inverse of subcritical class, then every maximal invariant subspace of T has codimension one, and if T has a finite dimensional kernel and a right inverse of subcritical class, then every minimal invariant subspace of T is one dimensional. Using these results we obtain new information on the invariant subspace lattices of shifts and backward shifts on a wide class of Banach spaces of analytic functions on the unit disc. We also introduce the notion of primary invariant subspaces, and determine their structure for a large class of shifts.

Original languageEnglish
Pages (from-to)155-213
Number of pages59
JournalJournal of Functional Analysis
Issue number1
StatePublished - 10 Sep 2001


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