Spectra of rank-one perturbations of self-adjoint operators

Direct and Inverse Spectral Problems for Rank-One Perturbations of Self-adjoint Operators

“…and thus can be analytically extended to σ 0 (A); we keep the notation F for this extension. As proved in [15], the geometric multiplicity of every eigenvalue μ of B is at most 2; multiplicity 2 is only possible when μ ∈ σ 0 (A) (i.e., when μ = λ n for some n ∈ I 0 ) and, in addition, a n = b n = F (λ n ) = 0. We also observe that when a n = b n = 0, then the subspace ls{v n } is invariant under both B and B * and thus is reducing for B. Denoting by H 0 the closed linear span of all such subspaces, we conclude that H 0 and H H 0 are reducing for B and the operators A and B coincide on H 0 .…”

“…In this section, we collect some properties of the rank-one perturbations of self-adjoint operators A acting in a fixed separable (infinite-dimensional) Hilbert space H established in [15] that will be used to prove the main results of this work.…”

“…This function appears in the Krein resolvent formula for B [9,15], and its zeros characterise the spectrum of B.…”

“…[2,9,37]), the question is quite non-trivial even for self-adjoint perturbations of a self-adjoint operator A, and thus necessarily much more complicated for generic rank-one perturbations. In our previous work [15], we described local spectral properties of rank-one perturbations of a self-adjoint operator with discrete spectrum and proved that the eigenvalues of B are asymptotically close to those of A at infinity. In particular, it was shown therein that such a perturbation can possess eigenvalues of arbitrarily prescribed multiplicities at any finite set of complex numbers thus significantly extending the finite-dimensional result of [23].…”

“…The structure of the paper is as follows. In the next section, we collect some basic spectral properties of the rank-one perturbations B and review the results of [15]. In Sect.…”