Spectral theory of rank one perturbations of normal compact operators

Abstract

We construct a functional model for rank one perturbations of compact normal operators acting in a certain Hilbert spaces of entire functions generalizing de Branges spaces. Using this model we study completeness and spectral synthesis problems for such perturbations. Previously, in [10] the spectral theory of rank one perturbations was developed in the selfadjoint case. In the present paper we extend and significantly simplify most of known results in the area. We also prove an Ordering Theorem for invariant subspaces with common spectral part. This result is essentially new even for rank one perturbations of compact selfadjoint operators.

Completeness Property of One-Dimensional Perturbations of Normal and Spectral Operators Generated by First Order Systems

“…In fact, this result was proved in [4,5] only for L 0 = L * 0 and was extended to the case of normal operators L 0 in a recent preprint by A. Baranov [3].…”

“…Theorem 1.5. [4,5,3] For any normal operator L 0 in H with simple point spectrum there exists peculiarly complete operator L such that the resolvent difference…”

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

“…On the side, we mention Baranov [12] where a model representation and a spectral synthesis for rank-one perturbations of normal operators is achieved.…”

Backward shift and nearly invariant subspaces of Fock-type spaces

“…In the context of Hardy spaces in general domains the equivalence of near invariance and division invariance is shown in [2, Proposition 5.1]; a similar argument works for general Banach spaces of analytic functions [6,Proposition 7.1].…”