Let D be a self-adjoint operator on a Hilbert space H and x a bounded
operator on H. We say that x is n-times weakly D-differentiable, if for any
pair of vectors a, b from H the function is n-times
differentiable. We give several characterizations of this property, among which
one is original. The results are used to show, that for a von Neumann algebra M
on H, the sub-algebra of n-times weakly D-differentiable operators has a
representation as a reflexive algebra of operators on a bigger Hilbert space.Comment: This version acknowledges results from the litterature, which the
first edition was unaware of. The result on the existence of a representation
with a reflexive image is ne
