Let E be a separable symmetric sequence space, and let CE be the unitary matrix space associated with E, i.e. the Banach space of all compact operators x on l2 so that s(x) E, with the norm , where are the s-numbers of x. One of the interesting subjects in the theory of the unitary matrix spaces is the clarification of correlation between the geometric properties of the spaces E and CE. A series of results in this direction related with the notions of type, cotype and uniform convexity of the spaces CE has been already obtained (see 13).
Abstract. We present several characterizations of Kadec-Klee properties in symmetric function spaces on the half-line, based on the K-functional of J. Peetre. In addition to the usual Kadec-Klee property, we study those symmetric spaces for which sequential convergence in measure (respectively, local convergence in measure) on the unit sphere coincides with norm convergence.
IntroductionIf (E, · E ) is a normed linear space, then E is said to have the Kadec-Klee property (sometimes called the Radon-Riesz property, or property (H)) if and only if sequential weak convergence on the unit sphere coincides with norm convergence. As is well known, this property was first studied by J. Radon [Ra] and subsequently by F. Riesz [Ri1], [Ri2] who showed that the classical L p -spaces, 1 < p < ∞, have the Kadec-Klee property. Although the space L 1 [0, 1] (with Lebesgue measure) fails to have the Kadec-Klee property, Riesz showed that each of the L p -spaces, 1 ≤ p < ∞, has the property that each sequence on the unit sphere that converges almost everywhere converges also in norm.In this paper, we study properties of Kadec-Klee type in the setting of symmetric (rearrangement-invariant) spaces E on the positive half-line in which sequential weak convergence is replaced by convergence in some natural linear space topology on E. In addition to the usual weak topology, we consider the weak topology induced on E by the ideal Λ ∞ = L 1 ∩L ∞ (which coincides with the weak topology if E has separable dual, and with the weak * -topology if E is the dual of some separable symmetric space) as well as the linear topologies of convergence in measure and local convergence in measure. Each of these linear topologies plays a natural and important role in the study of symmetric spaces. For example, it has been shown by Sedaev [Se1], [Se2] that each separable symmetric space on the positive half-line has an equivalent symmetric norm for which sequential σ(E, Λ ∞ ) convergence on the unit sphere coincides with norm convergence. This is to be compared with the well-known renorming theorem of Kadec [Ka1] that each separable Banach space admits an equivalent locally uniformly convex norm, and therefore has the Kadec-Klee property for some equivalent (Banach space) norm. The analogue and appropriate strengthening of the Kadec renorming theorem in the setting of general Banach lattices with order continuous norm has been given in [DGL], where it is also
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