Regular one-parameter groups, reflection positivity and their application to Hankel operators and standard subspaces

Standard subspaces are a well studied object in algebraic quantum field theory (AQFT). Given a standard subspace V of a Hilbert space H, one is interested in unitary one-parameter groups on H with Uₜ V ⊆ V for every t ∈ R₊. If ( V,U) is a non-degenerate standard pair on H, i.e. the self-adjoint infinitesimal generator of U is a positive operator with trivial kernel, two classical results are given by Borchers' Theorem, relating non-degenerate standard pairs to positive energy representations of the affine group Aff(R) and the Longo--Witten Theorem, stating the the semigroup of unitary endomorphisms of V can be identified with the semigroup of symmetric operator-valued inner functions on the upper half plane. In this thesis we prove results similar to the theorems of Borchers and of Longo--Witten for a more general framework of unitary one-parameter groups without the assumption that their infinitesimal generator is positive. We replace this assumption by the weaker assumption that the triple (H, V,U) is a so called real regular one-parameter group.