  We study heat and wave type equations on a separable Hilbert space
$\mathcal{H}$ by considering non-local operators in time with any positive
densely defined linear operator with discrete spectrum. We show the explicit
representation of the solution and analyse the time-decay rate in a scale of
suitable Sobolev space. We perform similar analysis on multi-term heat and
multi-wave type equations. The main tool here is the Fourier analysis which can
be developed in a separable Hilbert space based on the linear operator
involved. As an application, the same Cauchy problems are considered and
analysed in the setting of a graded Lie group. In this case our analysis relies
on the group Fourier analysis. An extra ingredient in this framework allows, in
the case of heat type equations, to establish $L^p$-$L^q$ estimates for
$1\leqslant p\leqslant 2\leqslant q<+\infty$ for the solutions on graded Lie
group groups. Examples and applications of the developed theory are given,
either in terms of self-adjoint operators on compact or non-compact manifolds,
or in the case of particular settings of graded Lie groups. The results of this
paper significantly extend in different directions the results of Part I, where
operators on compact Lie groups were considered. We note that the results
obtained in this paper are also new already in the Euclidean setting of
$\mathbb{R}^n$.
