Holomorphic Families of Subspaces of a Banach Space

Abstract

For a complex Banach space X let G(X) denote the set of all closed subspaces of X endowed with the gap-topology, see the definition below. Let Ω be a complex manifold. In this note we define and study holomorphic mappings form Ω into G(X). The class of those mappings will include (and actually consists of) all range and kernel families of holomorphic uniformly regular operator functions, i.e. operator functions A(z) which have locally the property that for all x e X, ‖ A(z)x ‖ #x2265; ηdist(x,N(A(z))) for some η > 0. This is in contrast to the holomorphic Banach vector bundles in the sense of [12] in general Banach spaces. An example will be given which shows that the category of holomorphic Banach vector bundles fails to have the “sum-intersection-property” (cf. Theorem 4) which turns out to be fundamental for our mappings: with the help of this property we prove that the sheaf of holomorphic sections of a holomorphic mapping into G(X) is a Banach coherent analytic Frechet sheaf in the sense of [5] which allows us to derive some global results; especially we solve the lifting problem of [3]. Some other concepts closely related to the theory presented here can be found in [8], [9] and [11] for example. There is however a gap in our theory: the complex manifold could clearly be replaced by some arbitrary reduced complex space but we do not know whether in this case Theorem 4 remains valid. A second problem is the following: is it possible to give G(X) the structure of an infinite dimensional complex space such that for every complex space Ω the holomorphic mappings from Ω into G(X) are the natural morphisms? Note that this is possible if we restrict ourselves to the open subset Gc(X) of all complemented subspaces of X (cf. [1]; unfortunately Gc(X) is not dense in G(X) in general; take e.g., X = l ∞; cf. the footnote on p.17 of [1]).