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.
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