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Invariant Hilbert spaces of analytic functions on bounded symmetric domains. (English) Zbl 0733.46011
Topics in operator theory. Ernst D. Hellinger Mem. Vol., Oper. Theory, Adv. Appl. 48, 67-91 (1990).
Summary: [For the entire collection see Zbl 0722.00019.]
We determine those Hilbert spaces of analytic functions on a bounded symmetric domain in \({\mathbb{C}}^ N\) which are mapped unitarily onto themselves by composition with each automorphism of the domain. We give several extensions of this result to cases when the action of the automorphism group is weighted or when the action is only isomorphic (and not isometric) or when the space is only a semi-Hilbert space; that is, the “norm” has a non-trivial kernel.

46E20 Hilbert spaces of continuous, differentiable or analytic functions