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Narrow operators (a survey)

Mikhail Popov (2011)

Banach Center Publications

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...

Narrow operators and rich subspaces of Banach spaces with the Daugavet property

Vladimir M. Kadets, Roman V. Shvidkoy, Dirk Werner (2001)

Studia Mathematica

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).

Near smoothness of Banach spaces.

Józef Banas, Kishin Sadarangani (1995)

Collectanea Mathematica

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

Ji Donghai, Yanming Lü, Ting Fu Wang (1998)

Commentationes Mathematicae Universitatis Carolinae

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.

New examples of K-monotone weighted Banach couples

Sergey V. Astashkin, Lech Maligranda, Konstantin E. Tikhomirov (2013)

Studia Mathematica

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 t 1 / p for some p ∈ [1,∞], then X = L p . At...

New extension of the variational McShane integral of vector-valued functions

Sokol Bush Kaliaj (2019)

Mathematica Bohemica

We define the Hake-variational McShane integral of Banach space valued functions defined on an open and bounded subset G of m -dimensional Euclidean space m . It is a “natural” extension of the variational McShane integral (the strong McShane integral) from m -dimensional closed non-degenerate intervals to open and bounded subsets of m . 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...

Nonatomic Lipschitz spaces

Nik Weaver (1995)

Studia Mathematica

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.

Non-commutative martingale VMO-spaces

Narcisse Randrianantoanina (2009)

Studia Mathematica

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).

Noncommutative Valdivia compacta

Marek Cúth (2014)

Commentationes Mathematicae Universitatis Carolinae

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 K , we show that K is Corson if and only if every continuous image...

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