  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.
