A measure-theoretical approach to the nuclear and inductive spectral theorems

The usual mathematical implementations for the generalized eigenvectors and eigenfunctions of a spectral measure (or a normal operator) on a Hilbert space H use direct integral decompositions of the space H or auxiliary subspaces Φ with their topology τΦ, so that the generalized eigenvectors belong to the components of the direct integral or to the (anti)dual space Φ×, respectively. In this work the Gelfand-Vilenkin description of the generalized eigenvectors, in terms of certain Radon-Nikodym derivatives associated to the spectral measure, permit us to give new proofs of renewed inductive and nuclear versions of the spectral theorem, casting new insight on the measure-theoretical nature of these results.