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We present a spectral-theoretic approach to time-average statistical mechanics for general, nonequilibrium initial conditions. We consider the statistics of bounded, local additive functionals of reversible as well as irreversible ergodic stochastic dynamics with continuous or discrete state-space. We derive exact results for the mean, fluctuations, and correlations of time-average observables from the eigenspectrum of the underlying generator of Fokker-Planck or master equation dynamics, and we discuss the results from a physical perspective. Feynman-Kac formulas are rederived using Itô calculus and combined with non-Hermitian perturbation theory. The emergence of the universal central limit law in a spectral representation is shown explicitly on large-deviation timescales. For reversible dynamics with equilibrated initial conditions, we derive a general upper bound to fluctuations of occupation measures in terms …