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Holomorphic automorphisms of the unit ball of a direct sum. (English) Zbl 0747.46040
If \(B_ E\) is the unit ball of a Banach space \(E\) then there exists a Banach subspace \(\tilde E\) of \(E\) such that the orbit of the origin under the group of biholomorphic automorphisms of \(B_ E\) is equal to \(B_ E\cap\tilde E\) [W. Kaup-H. Upmeier, Proc. Am. Math. Soc. 58, 129- 133 (1976; Zbl 0337.32012)]. If \(E\) is an \(L^ p\) space then \(\tilde E=E\) if \(p=2\) or \(\infty\) and \(\tilde E=\{0\}\) if \(p\neq 2\) and \(p\neq\infty\) [E. Vesentini, Several complex variables, Proc. Int. Conf. Cortona/Italy 1976-1977, 282-284 (1978; Zbl 0443.32016) and R. Braun-W. Kaup-H. Upmeier, Manuscr. Math. 25, 97-133 (1978; Zbl 0398.32001)].
The author considers biholomorphic automorphisms of the unit ball of spaces of the form \(F\oplus_ pG\) where \(F\) and \(G\) are Banach spaces (and also somewhat more general spaces) and proves that \[ F\widetilde{\oplus_ p}G=\{0\}\text{ if } p\neq 2,\infty\text{ and } F\widetilde{\oplus_ \infty}G=F\oplus_ \infty G \] and shows that the case \(p=2\) is rather more complicated.
Reviewer: S.Dineen

46G20 Infinite-dimensional holomorphy
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