The recursion method of Haydock, Heine and Kelly is a powerful tool for calculating diagonal matrix elements of the resolvent of quantum-mechanical Hamiltonian operators by elegantly expressing them in terms of continued fractions. In this paper we extend the recursion method to off-diagonal matrix elements of general (possibly non-Hermitian) operators and apply it to the simulation of molecular optical absorption and photoemission spectra within time-dependent density-functional and many-body perturbation theories, respectively. This method is demonstrated with a couple of applications to the optical absorption and photoemission spectra of the caffeine molecule.