We study asymptotic quasinormal mode expansions of linear fields propagating on a black hole background by adopting a Keldysh scheme for the spectral construction of the resonant expansions. This scheme requires to cast quasinormal modes in terms of a non-selfadjoint problem, something achieved by adopting a hyperboloidal scheme for black hole perturbations. The method provides a spectral version of Lax-Phillips resonant expansions, adapted to the hyperboloidal framework, and extends and generalises Ansorg & Macedo black hole quasinormal mode expansions beyond one-dimensional problems. We clarify the role of scalar product structures in the Keldysh construction that prove non-necessary to construct the resonant expansion, in particular providing a unique quasinormal mode time-series at null infinity, but are required to define constant coefficients in the bulk resonant expansion by introducing a notion of 'size' (norm). By (numerical) comparison with the time-domain signal for test-bed initial data, we demonstrate the efficiency and accuracy of the Keldysh spectral approach. Indeed, we are able to recover Schwarzschild black hole tails, something that goes beyond the a priori limits of validity of the method and constitutes one of the main results. We also demonstrate the critical role of highly-damped quasinormal mode overtones to accurately account for the early time behaviour. As a by-product of the analysis, Sobolev H^p pseudospectra are constructed and the convergence issues of the black hole quasinormal mode pseudospectra are clarified in agreement with Warnick’s theorem.