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For a linear operator T in a Banach space let denote the point spectrum of T, let for finite n > 0 be the set of all such that dim ker(T - λ) = n and let be the set of all for which ker(T - λ) is infinite-dimensional. It is shown that is , is and for each finite n the set is the intersection of an set and a set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more detailed decomposition...
We prove that every (extended) Borel subset E of X × Y, where X is complete metric and Y is Polish, can be covered by countably many extended Borel sets with compact sections if the sections , x ∈ X, are σ-compact. This is a nonseparable version of a theorem of Saint Raymond. As a by-product, we get a proof of Saint Raymond’s result which does not use transfinite induction.
Conditions are given which enable or disable a complex space to be mapped biholomorphically onto a bounded closed analytic subset of a Banach space. They involve on the one hand the Radon-Nikodym property and on the other hand the completeness of the Caratheodory metric of .
A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/ hull to every negligible/measurable subset of [0,1]?
Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent....
We construct a Choquet simplex whose set of extreme points is -analytic, but is not a -Borel set. The set has the surprising property of being a set in its Stone-Cech compactification. It is hence an example of a set that is not absolute.
On cherche à donner une construction aussi simple que possible d'un borélien donné d'un produit de deux espaces polonais. D'où l'introduction de la notion de classe de Wadge potentielle. On étudie notamment ce que signifie "ne pas être potentiellement fermé", en montrant des résultats de type Hurewicz. Ceci nous amène naturellement à des théorèmes d'uniformisation partielle, sur des parties "grosses", au sens du cardinal ou de la catégorie.
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