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Standard Subspaces of Hilbert Spaces of Holomorphic Functions on Tube Domains

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  • Karl Hermann Neeb, Friedrich-Alexander University Erlangen-Nürnberg
  • ,
  • Bent Ørsted
  • Gestur Ólafsson, Louisiana State University

In this article we study standard subspaces of Hilbert spaces of vector-valued holomorphic functions on tube domains E+ iC, where C⊆ E is a pointed generating cone invariant under eRh 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, eπih) 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.

TidsskriftCommunications in Mathematical Physics
Sider (fra-til)1437-1487
Antal sider51
StatusUdgivet - sep. 2021

Bibliografisk note

Funding Information:
The research of K.-H. Neeb was partially supported by DFG-Grant Ne 413/10-1. The research of G. Ólafsson was partially supported by Simons Grant 586106.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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