A mathematical interpretation of Dirac's formalism. Part B: Generalized eigenfunctions in trajectory spaces

Starting with a Hilbert space we introduce the dense subspace where is a positive self-adjoint Hilbert–Schmidt operator on 2(R,µ). For the space a measure-theoretical Sobolev lemma is proved. The results for the spaces of type are applied to nuclear analyticity spaces , where e-t is a Hilbert–Schmidt operator on the Hilbert space X for each t>0. We solve the so-called generalized eigenvalue problem for a general self-adjoint operator in X.