Nonunitary similarity transformation of conservative to dissipative evolutions: Intertwining without time operator

Abstract

Reversible evolutions are usually expressed in terms of unitary groups on separable Hilbert spaces, whereas irreversible ones are described by contraction semigroups. In the theory of nonunitary similarity transformations intertwining unitary groups and contraction semigroups, proposed initially in the context of statistical mechanics as part of an exact theory of irreversibility, the unitary groups with such intertwining property have been qualified by the existence of an internal time operator. This work tackles the question of existence of internal time operators for unitary groups with the intertwining property. Equivalent conditions to the existence of internal time operators for such unitary groups are given on the basis of the Sz.-Nagy–Foiaş [Harmonic Analysis of Operators on Hilbert Spaces (North-Holland, Amsterdam, 1970)] dilation theory and the theory of shift invariant subspaces. These conditions permit us to solve the inverse intertwining problem in the negative: there are unitary groups with the intertwining property which do not admit internal time operator. A representative family of such unitary groups is given.