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On the holomorphic rigidity of linear operators on complex Banach spaces. (English) Zbl 0964.46026
The main notion in this paper is the following one. Let \(U,V\) be domains in complex Banach spaces. A holomorphic mapping \(f:U \to V\) is \(m\)-rigid at the point \(a \in U\) if for any holomorphic mapping \(g:U \to V\) it follows from \(f^{(k)}(a) = g^{(k)}(a)\) for all \(k=0,\dots,m-1\) that \(f=g\). In section 2 the authors present the basic background for holomorphic mappings and basic properties of \(m\)-rigid mappings. Section 3 is devoted the study of rigid linear operators. In section 4 the authors introduce numerical invariants \(\alpha_m\) to measure the non-rigidity of a linear operator. Moreover, they also define invariants \(\pi_m\) to measure certain eccentricities. In section 5 the invariants \(\alpha_m\) are determined and the \(\pi_m's\) are estimated for some special examples. Relations between rigidity and smoothness properties of the unit spheres are studied for contractive projections. In section 6 the authors introduce various types of tangent spaces and correlate them to the rigidity problem. In section 7 these methods are applied to JB\(^{\ast}\)-triples. The main result – Theorem 7.14 – solves completely the rigidity problem for \(w^{\ast}\)-closed inner ideals in JBW\(^{\ast}\)-triples.

MSC:
46G20 Infinite-dimensional holomorphy
58C10 Holomorphic maps on manifolds
47L07 Convex sets and cones of operators
46L70 Nonassociative selfadjoint operator algebras
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