Standard Subspaces of Hilbert Spaces of Holomorphic Functions on Tube Domains
Karl-Hermann Neeb, Bent Ørsted, Gestur Ólafsson
Abstract
In this article we study standard subspaces of Hilbert spaces of vector-valued holomorphic functions on tube domains $$E + i C^0$$ , where $$C \subseteq E$$ is a pointed generating cone invariant under $$e^{{{\mathbb {R}}}h}$$ for some endomorphism $$h \in \mathop {\mathrm{End}}\nolimits (E)$$ , diagonalizable with the eigenvalues $$1,0,-1$$ (generalizing a Lorentz boost). This data specifies a wedge domain $$W(E,C,h) \subseteq E$$ and one of our main results exhibits corresponding standard subspaces as being generated using test functions on these domains. We also investigate aspects of reflection positivity for the triple $$(E,C,e^{\pi i h})$$ and the support properties of distributions on E, arising as Fourier transforms of operator-valued measures defining the Hilbert spaces $$\mathcal {H}$$ . For the imaginary part of these distributions, we find similarities to the well known Huygens’ principle, relating to wedge duality in the Minkowski context. Interesting examples are the Riesz distributions associated to euclidean Jordan algebras.