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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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