Commutator estimates and R-flows in non-commutative operator spaces

The principal results in this paper are concerned with the description of domains of infinitesimal generators of strongly continuous groups of isometries in non-commutative operator spaces E(M, T), which are induced by R-flows on M. In particular, we are concerned with the description of operator functions which leave the domain of such generators invariant in all symmetric operator spaces, associated with a semi-finite von Neumann algebra M and a separable function space E on (0, infinity). Furthermore, we apply our results to the study of operator functions for which [D, x] E E(M, T) implies that [D, f(x)] E E(M, T), where D is an unbounded self-adjoint operator. Our methods are partly based on the recently developed theory of double operator integrals in symmetric operator spaces and the theory of adjoint C(0)-semigroups.