  We present an expansion of a many-body correlation function in a sum of
pseudomodes - exponents with complex frequencies that encompass both decay and
oscillations. We demonstrate that, typically, it is enough to take a few first
terms of this infinite sum to obtain an excellent approximation to the
correlation function at any time, with the large time behavior being determined
solely by the first pseudomode. The pseudomode expansion emerges in the
framework of the Heisenberg version of the recursion method. This method
essentially solves Heisenberg equations in a Lanczos tridiagonal basis
constructed in the Krylov space of a given observable. To obtain pseudomodes,
we first add artificial dissipation satisfying the dissipative generalization
of the universal operator growth hypothesis, and then take the limit of the
vanishing dissipation strength. Fast convergence of the pseudomode expansion is
facilitated by the localization in the Krylov space, which is generic in the
presence of dissipation and can survive the limit of the vanishing dissipation
strength. As an illustration, we apply the pseudomode expansion to calculate
infinite-temperature autocorrelation functions in the quantum Ising and $XX$
spin-$1/2$ models on the square lattice.
