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Journal of Operator Theory

Volume 85, Issue 1, Winter 2021  pp. 45-78.

Spectral dissection of finite rank perturbations of normal operators

Authors:  Mihai Putinar (1), Dmitry Yakubovich (2)
Author institution: (1) University of California at Santa Barbara, CA U.S.A. and Newcastle University, Newcastle upon Tyne, U.K.
(2) Departamento de Matematicas, Universidad Autonoma de Madrid, Spain and ICMAT (CSIC - UAM - UC3M - UCM), Cantoblanco 28049, Spain

Summary: Finite rank perturbations $T=N+K$ of a bounded normal operator $N$ acting on a separable Hilbert space are studied thanks to a natural functional model of $T$; in its turn the functional model solely relies on a perturbation matrix/characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of $T$. Under mild geometric conditions on the spectral measure of $N$ and some smoothness constraints on $K$ we show that the operator $T$ admits invariant subspaces, or even it is decomposable.

Keywords: normal operator, perturbation determinant, Cauchy transform, decomposable operator, functional model, Bishop's property $(\beta)$

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