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Holomorphic retracts in \(B_H^{\infty}\). (English) Zbl 1091.37010
M. Budzynska and T. Kuczumow [Contemporary Mathematics 364, 35–40 (2004; Zbl 1091.58005)] proved that if \(\mathcal {F}\) is a countable family of holomorphic (\(k_{ B_H^{\infty}}\)-nonexpansive) commuting self-mappings of \(B_H^{\infty}\) with a nonempty common fixed-point set, then this fixed-point set is a \(k_{ B_H^{\infty}}\)-nonexpansive retract of \(B_H^{\infty}\).
In the present paper, the authors prove a general result of the same type: they show that the common fixed-point set of a commuting family of holomorphic mappings in \(B_H^{\infty}\) is either empty or a holomorphic set.
MSC:
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
46T25 Holomorphic maps in nonlinear functional analysis
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
32A10 Holomorphic functions of several complex variables
58B12 Questions of holomorphy and infinite-dimensional manifolds
Citations:
Zbl 1091.58005
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References:
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