  We consider stochastic semi-linear evolution equations which are driven by
additive, spatially correlated, Wiener noise, and in particular consider
problems of heat equation (analytic semigroup) and damped-driven wave equations
(bounded semigroup) type. We discretize these equations by means of a spectral
Galerkin projection, and we study the approximation of the probability
distribution of the trajectories: test functions are regular, but depend on the
values of the process on the interval $[0,T]$.
  We introduce a new approach in the context of quantative weak error analysis
for discretization of SPDEs. The weak error is formulated using a deterministic
function (It\^o map) of the stochastic convolution found when the nonlinear
term is dropped. The regularity properties of the It\^o map are exploited, and
in particular second-order Taylor expansions employed, to transfer the error
from spectral approximation of the stochastic convolution into the weak error
of interest.
  We prove that the weak rate of convergence is twice the strong rate of
convergence in two situations. First, we assume that the covariance operator
commutes with the generator of the semigroup: the first order term in the weak
error expansion cancels out thanks to an independence property. Second, we
remove the commuting assumption, and extend the previous result, thanks to the
analysis of a new error term depending on a commutator.
