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Convexity and fixed points of holomorphic mappings in Hilbert ball and polydisc. (English) Zbl 0606.47057

Let B be the open unit ball in a complex Hilbert space. For \(x_ 1,x_ 2\in B\) define \(\sigma (x_ 1,x_ 2):=(1-\| x_ 1\|^ 2)(1- \| x_ 2\|^ 2)/| 1-(x_ 1,x_ 2)|^ 2\). The Carathéodory metric on B is then given by \(\rho_ 1(x,y):=\tanh^{- 1}(1-\sigma (x,y))^{1/2}\). Call \(B^ n\) the product of n copies of B and define recursively \(\rho_ n(x_ 1,y_ 1),(x_ 2,y_ 2)):=\max \{\rho_ 1(x_ 1,x_ 1),\rho_{n-1}(y_ 1,y_ 2)\}\) for \((x_ 1,x_ 2)\in B\), \((y_ 1,y_ 2)\in B^{n-1}\). By exploiting the fact that a holomorphic map \(T: B^ n\to B^ n\) is nonexpansive with respect to \(\rho_ n\) the authors obtain the following results:
If \(T: B^ 2\to B^ 2\) is holomorphic with nonempty fixed point set F then either F is a single point or the weak closure of F meets the boundary of \(B^ 2\). Moreover, \(F\neq \emptyset\) if and only if there exists a nonempty convex set \(X\subset B^ 2\) with TX\(\subset X\) such that the weak closure of TX is contained in \(B^ 2.\)
An easier criterion for the existence of fixed points is given by the following theorem:
Let \(T: B^ 2\to \bar B^ 2\) be a holomorphic map and assume that T admits a continuous extension over \(\bar B{}^ 2\), then T has a fixed point in \(\bar B{}^ 2.\)
It may seem remarkable that it is unknown whether one may replace 2 by \(n\geq 3\) in this result.
Reviewer: Chr.Fenske

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
46G20 Infinite-dimensional holomorphy
58C10 Holomorphic maps on manifolds
46A55 Convex sets in topological linear spaces; Choquet theory
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