Spectral Analysis, Model Theory and Applications of Finite-Rank Perturbations

Abstract

This survey focuses on two main types of finite-rank perturbations: self-adjoint and unitary. We describe both classical and more recent spectral results. We pay special attention to singular self-adjoint perturbations and model representations of unitary perturbations.

Spectral dissection of finite rank perturbations of normal operators

“…If (D ′ ) holds and the measure µ satisfies an estimate µ(Q) ≤ C|Q| σ for all squares Q, where σ ∈ (1, 2] and C are constants, then Ψ is a Hölderα function for some positive α. This is shown along the lines of the last part of the proof of Lemma 2.3, using the estimate (21) with some minor modifications. We leave the details to the reader.…”

“…Let w ∈ C \ G s (ν), |z − w| < δ, where δ is to be determined. Choose r as above, and assume δ < r. Then (21)…”

“…It is sufficient to mention the case of dissipative operators with rank-one imaginary part or the celebrated phase-shift and related perturbation determinant, see [22] for the golden era references, or [44] for more recent developments. The booming topics of Aleksandrov-Clark measures [38] and the resurrection of Aronszjan-Donoghue theory for matrix valued measures associated to finite rank perturbations of self-adjoint operators [33,21] are two other notable examples. To name only one less known, additional relevant ramification: an apparently non-related open problem of approximation theory, known as Sendov conjecture, can be translated into spectral estimates of rank-two perturbations of normal matrices, see [28].…”