Koopman's approach to dissipation

We analyze spectral properties of Koopman's operator corresponding to classical dissipative dynamics. It turns out that apart from the real continuous spectrum such operator gives rise to the family of discrete complex generalized eigenvalues. The corresponding generalized eigenvectors are interpreted as resonant states. We show that Koopman's operator may be entirely characterized in terms of resonant states and the main message is that resonant states are responsible for dissipation.