Standard Subspaces of Hilbert Spaces of Holomorphic Functions on Tube Domains

• Published in: Communications in mathematical physics 2021-09, Vol.386 (3), p.1437-1487
• Main Authors: Neeb, Karl-Hermann, Ørsted, Bent, Ólafsson, Gestur
• Format: Article
• Language: English
• Subjects: Analytic functions; Classical and Quantum Gravitation; Complex Systems; Domains; Eigenvalues; Fourier transforms; Hilbert space; Mathematical analysis; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Subspaces; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-021-04144-5

In this article we study standard subspaces of Hilbert spaces of vector-valued holomorphic functions on tube domains E + i C 0 , where C ⊆ E is a pointed generating cone invariant under e R h for some endomorphism h ∈ End ( E ) , diagonalizable with the eigenvalues 1 , 0 , - 1 (generalizing a Lorentz boost). This data specifies a wedge domain W ( E , C , h ) ⊆ 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 π i h ) and the support properties of distributions on E , arising as Fourier transforms of operator-valued measures defining the Hilbert spaces  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.