The Lomonosov Theorem

Carlos S. Kubrusly
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Carlos S. Kubrusly: Catholic University of Rio de Janeiro

Chapter 12 in Hilbert Space Operators, 2003, pp 129-142 from Springer

Abstract: Abstract Compact operators (on a complex Banach space of dimension greater than 1) have a nontrivial invariant subspace; a nontrivial hyperinvariant subspace, actually, if it is nonscalar. The definitive result in this line is due to Lomonosov [40]: An operator has a nontrivial invariant subspace if it commutes with a nonscalar operator that commutes with a nonzero compact operator. In fact, every nonscalar operator that commutes with a nonscalar compact operator (itself, in particular) has a nontrivial hyperinvariant subspace. Recall that on an infinite-dimensional formed space the only scalar compact operator is the null operator; on a finite-dimensional formed space every operator is compact.

Keywords: Invariant Subspace; Compact Operator; Nonzero Vector; Complex Hilbert Space; Complex Banach Space (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-1-4612-2064-0_12

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