# Spectral dissection of finite rank perturbations of normal operators

## TL;DR

This paper investigates how finite rank perturbations affect the spectral properties of normal operators using a functional model based on a perturbation matrix, revealing conditions for invariant subspaces and decomposability.

## Contribution

It introduces a functional model for finite rank perturbations of normal operators that encodes spectral behavior via a perturbation matrix, providing new insights into invariant subspaces.

## Key findings

- Operator T admits invariant subspaces under certain conditions.
- Operator T can be decomposable with mild spectral measure assumptions.
- Functional model links spectral behavior to a perturbation matrix.

## Abstract

Finite rank perturbations $T=N+K$ of a bounded normal operator $N$ 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.