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Narrow operators are those operators defined on function spaces which are "small" at signs, i.e., at {-1,0,1}-valued functions. We summarize here some results and problems on them. One of the most interesting things is that if E has an unconditional basis then each operator on E is a sum of two narrow operators, while the sum of two narrow operators on L₁ is narrow. Recently this notion was generalized to vector lattices. This generalization explained the phenomena of sums: the set of all regular...
Let X be a Banach space. We introduce a formal approach which seems to be useful in the study of those properties of operators on X which depend only on the norms of the images of elements. This approach is applied to the Daugavet equation for norms of operators; in particular we develop a general theory of narrow operators and rich subspaces of spaces X with the Daugavet property previously studied in the context of the classical spaces C(K) and L₁(μ).
The aim of this paper is to discuss the concept of near smoothness in some Banach sequence spaces.
Nearly smooth points and near smoothness in Orlicz spaces are characterized. It is worth to notice that in the nonatomic case smooth points and nearly smooth points are the same, but in the sequence case they differ.
A necessary condition for Kostyuchenko type systems and system of powers to be a basis in (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in (1 ≤ p < +∞).
Some new examples of K-monotone couples of the type (X,X(w)), where X is a symmetric space on [0,1] and w is a weight on [0,1], are presented. Based on the property of w-decomposability of a symmetric space we show that, if a weight w changes sufficiently fast, all symmetric spaces X with non-trivial Boyd indices such that the Banach couple (X,X(w)) is K-monotone belong to the class of ultrasymmetric Orlicz spaces. If, in addition, the fundamental function of X is for some p ∈ [1,∞], then . At...
We define the Hake-variational McShane integral of Banach space valued functions defined on an open and bounded subset of -dimensional Euclidean space . It is a “natural” extension of the variational McShane integral (the strong McShane integral) from -dimensional closed non-degenerate intervals to open and bounded subsets of . We will show a theorem that characterizes the Hake-variational McShane integral in terms of the variational McShane integral. This theorem reduces the study of our...
We abstractly characterize Lipschitz spaces in terms of having a lattice-complete unit ball and a separating family of pure normal states. We then formulate a notion of "measurable metric space" and characterize the corresponding Lipschitz spaces in terms of having a lattice complete unit ball and a separating family of normal states.
We study Banach space properties of non-commutative martingale VMO-spaces associated with general von Neumann algebras. More precisely, we obtain a version of the classical Kadets-Pełczyński dichotomy theorem for subspaces of non-commutative martingale VMO-spaces. As application we prove that if ℳ is hyperfinite then the non-commutative martingale VMO-space associated with a filtration of finite-dimensional von Neumannn subalgebras of ℳ has property (u).
We prove some generalizations of results
concerning Valdivia compact spaces
(equivalently spaces with a commutative
retractional skeleton) to the spaces
with a retractional skeleton
(not necessarily commutative).
Namely, we show that the dual unit ball
of a Banach space is Corson provided
the dual unit ball of every equivalent
norm has a retractional skeleton.
Another result to be mentioned is the
following. Having a compact space ,
we show that is Corson if and only
if every continuous image...
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