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Completely Indecomposable Operators and a Uniqueness Theorem of Cartwright–Levinson Type

✍ Scribed by A. Atzmon; M. Sodin

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 224 KB
Volume: 169
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1999.3454

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✦ Synopsis

A bounded linear operator T on a complex Hilbert space will be called completely indecomposable if its spectrum is not a singleton, and is included in the spectrum of the restrictions of T and T * to any of their nonzero invariant subspaces. Two classes of completely indecomposable operators are constructed. The first consists of essentially selfadjoint operators with spectrum [&2, 2], and the second of bilateral weighted shifts whose spectrum is the unit circle. We do not know whether any of the operators in the first class has a proper invariant subspace and if any of the operators in the second class has a proper hyperinvariant subspace. We also establish a new uniqueness theorem of Cartwright Levinson type which is the main ingredient in our proofs of complete indecomposability.

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Completely Indecomposable Operators and

Completely Indecomposable Operators and a Uniqueness Theorem of Cartwright–Levinson Type: Addendum to

✍ A. Atzmon; M. Sodin 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 72 KB

The purpose of this note is to show that the answers to Problems 2, 3, and 5 in [1] are positive, even under weaker assumptions. Keeping the notations of [1], we have Theorem 1. If : is a positive sequence on Z + which converges to a nonzero limit, then the operator A(:) has a proper invariant subsp