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Rank-one perturbations of diagonal operators. (English) Zbl 0979.47012

Summary: We study rank-one perturbations of diagonal Hilbert space operators mainly from the standpoint of invariant subspace problem. In addition to proving some general properties of these operators, we identify the normal operators and contractions in this class. We show that two well known results about the eigenvalues of rank-one perturbations and one-codimension compressions of selfadjoint compact operators are equivalent. Sufficient conditions are given for existence of nontrivial invariant subspaces for this class of operators.

MSC:

47A55 Perturbation theory of linear operators
47A15 Invariant subspaces of linear operators
30B50 Dirichlet series, exponential series and other series in one complex variable
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B07 Linear operators defined by compactness properties
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