Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum
Let G/H be a semisimple symmetric space. Then the space L2 (G/H) can be decomposed into a finite sum of series of representations induced from parabolic subgroups of G. The most continuous part of the spectrum of L 2 (G/H) is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions on a bounded symmetric domain of tube type, containing G/H as a boundary component. A boundary value map is constructed and we show that it induces a G-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part Lmc2(G/H) of L2(G/H). We also relate our Hardy space to the classical Hardy space on the bounded symmetric domain.
Publication Source (Journal or Book title)
Gindikin, S., Krötz, B., & Ólafsson, G. (2003). Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum. Mathematische Annalen, 327 (1), 25-66. https://doi.org/10.1007/s00208-003-0409-x