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Measures of weak non-compactness in spaces of nuclear operators. (English) Zbl 1421.46011

There are several ways of measuring weak noncompactness of subsets of Banach spaces. Let \(X\) be a Banach space and \(A, B \subset X\) two nonempty sets. Define \[ \hat{d}(A,B) = \sup\{ \mathrm{dist}(a,B) : a \in A\}. \] The Hausdorff measure of noncompactness of a bounded set \(A \subset X\) is given by \[ \chi(A) = \inf\{ \hat{d}(A,K) : K \subset X \ \text{compact}\}. \] A natural generalization to weak noncompactness is De Blasi’s measure of weak noncompactness \[ \omega(A) = \inf\{ \hat{d}(A,K) : K \subset X \text{ weakly compact}\}. \] Another measure of weak noncompactness is the following, which is inspired by the Banach-Alaoğlu theorem, \[ \mathrm{wk}_X(A) = \hat{d}(\overline{A}^{w^*},X). \] Here, we consider \(X\) canonically embedded in its bidual and take the weak\(^*\)-closure also in the bidual. There is yet another measure of weak noncompactness inspired by the Eberlein-Šmulyan theorem considered in the paper and it is equivalent to \(\mathrm{wk}_X(\cdot)\).
The quantities \(\mathrm{wk}_X(\cdot)\) and \(\omega(\cdot)\) are, in general, not equivalent, but for many classical Banach spaces they coincide, e.g., if \(X = L_1(\mu)\) or \(X = c_0(\Gamma)\). Problem 1 of this paper asks if there is a classical Banach space \(X\) in which the measures \(\mathrm{wk}_X(\cdot)\) and \(\omega(\cdot)\) are not equivalent.
The authors show that, if \(\Lambda\) and \(J\) are infinite sets and \(p,q \in (1,\infty)\), then \(\mathrm{wk}_X(A) = \omega(A)\) for all bounded sets \(A \subset X := N(\ell_q(\Lambda),\ell_p(J))\) (the space of nuclear operators). It is also shown that, if \(M\) is an atomic von Neumann algebra, then \(\mathrm{wk}_X(\cdot)\) and \(\omega(\cdot)\) coincide on bounded subsets of the predual of \(M\).

MSC:

46B04 Isometric theory of Banach spaces
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
46B50 Compactness in Banach (or normed) spaces
46B28 Spaces of operators; tensor products; approximation properties
46L10 General theory of von Neumann algebras
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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