Polynomial numerical hull equals the polynomial convex hull of the spectrum of an operator and it can in principle be computed if the norms of polynomials of the operator are available.
It was noticed that it is more natural to ask for a concrete representation for the resolvent, outside the polynomial numerical hull, rather than trying to compute the set in which such a representation does not exist. This lead to a surprisingly simple general representation of the resolvent operator as a power series, in which the speed of convergence is associated with a Green's function for the outside of the polynomial numerical hull. The representation opens a lot of new questions. The aim is to study in particular the behaviour of the resolvent under analytic perturbations of the operator.