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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.

In resonance to a recent geometric framework proposed by Douglas and Yang, afunctional model for certain linear bounded operators with rank-oneself-commutator acting on a Hilbert space is developed. By taking advantage ofthe refined existing theory of the principal function of a hyponormal operatorwe transfer the whole action outside the spectrum, on the resolvent of theunderlying operator, localized at a distinguished vector. The wholeconstruction turns out to rely on an elementary algebra body involving analyticmultipliers and Cauchy transforms. A natural field theory interpretation of theresulting resolvent functional model is proposed.