  For classical dynamical systems time operators are introduced as selfadjoint
operators satisfying the so called weak Weyl relation with the unitary groups
of time evolution. Dynamical systems with time operators are intrinsically
irreversible because they admit Lyapounov operators as functions of the time
operator. For quantum systems selfadjoint time operators are defined in the
same way or as dilations of symmetric ones dealing with the time-energy
uncertainty relation, times of occurrence or survival probabilities. This work
tackles the question of the existence of selfadjoint time operators on the
basis of the Halmos-Helson theory of invariant subspaces, the Sz.-Nagy-Foias
dilation theory and the Misra-Prigogine-Courbage theory of irreversibility. It
is shown that the existence of a selfadjoint time operator for a unitary
evolution is equivalent to the intrinsic irreversibility of the evolution plus
the existence of a simply invariant subspace or a rigid operator-valued
function for its Sz.-Nagy-Foias functional model. An extensive set of
equivalent conditions to the existence of selfadjoint time operators can be
obtained from these results. Such conditions are written in terms of
Schr\"odinger couples, the Weyl commutation relation, incoming and outgoing
subspaces, innovation processes, Lax-Phillips scattering processes, and
translation and spectral representations. As an example, the selfadjoint
extension of the quantum Aharonov-Bohm time-of-arrival operator is studied.
