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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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