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Semigroups of holomorphic isometries. (English) Zbl 0642.47035
Let B be the open unit ball of a complex Hilbert space \({\mathcal H}\). Let \({\mathcal H}\oplus {\mathbb{C}}\) be the Hilbert space direct sum of \({\mathcal H}\) and \({\mathbb{C}}\), with inner product (,), and let \(\alpha\) be the continuous hermitian sesquilinear form defined by \(\alpha (p,q)=(Jp,q)\) where p,q\(\in {\mathcal H}\oplus {\mathbb{C}}\), J is the operator \(J=I\oplus (-1)\) and I is the identity on \({\mathcal H}\). \(G_ 0\) is the group of all continuous invertible linear operators in \({\mathcal H}\oplus {\mathbb{C}}\) leaving the form \(\alpha\) invariant. G is the semigroup G of all continuous linear operators in \({\mathcal H}\oplus {\mathbb{C}}\) leaving the form \(\alpha\) invariant. Aut B is the group of all holomorphic automorphisms of B. Iso B is the semigroup of all holomorphic maps \(B\to B\) which are isometries for the hyperbolic differential metric of B.
The main result is the following:
Theorem. There is a surjective homomorphism of G onto Iso B, mapping \(G_ 0\) onto Aut B, whose kernel is the center of G. The strongly continuous linear semigroup \(T:R_+\to G\) is characterized by use of their infinitesimal generator X. The Cauchy problem associated with the infinitesimal generator X of the semigroup T is presented.
Reviewer: Wu Liangsen

MSC:
47D03 Groups and semigroups of linear operators
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