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On Some Classes of Operators on C(K,X)

Ioana Ghenciu (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ is the σ-algebra of Borel subsets of K, C(K,X) is the Banach space of all continuous X-valued functions (with the supremum norm), and T:C(K,X) → Y is a strongly bounded operator with representing measure m:Σ → L(X,Y). We show that if T is a strongly bounded operator and T̂:B(K,X) → Y is its extension, then T is limited if and only if its extension T̂ is limited, and that T* is completely continuous (resp. unconditionally...

On some convexity properties in the Besicovitch-Musielak-Orlicz space of almost periodic functions with Luxemburg norm

Fazia Bedouhene, Amina Daoui, Hocine Kourat (2012)

Commentationes Mathematicae Universitatis Carolinae

In this article, it is shown that geometrical properties such as local uniform convexity, mid point local uniform convexity, H-property and uniform convexity in every direction are equivalent in the Besicovitch-Musielak-Orlicz space of almost periodic functions ( B ˜ ϕ a . p . ) endowed with the Luxemburg norm.

On some equivalent geometric properties in the Besicovitch-Orlicz space of almost periodic functions with Luxemburg norm

Fazia Bedouhene, Mohamed Morsli, Mannal Smaali (2010)

Commentationes Mathematicae Universitatis Carolinae

The paper is concerned with the characterization and comparison of some local geometric properties of the Besicovitch-Orlicz space of almost periodic functions. Namely, it is shown that local uniform convexity, H -property and strict convexity are all equivalent. In our approach, we first prove some metric type properties for the modular function associated to our space. These are then used to prove our main equivalence result.

On some geometric properties concerning closed convex sets.

D. N. Kutzarova, Pei-Kee Lin, P. L. Papini, Xin Tai Yu (1991)

Collectanea Mathematica

In this article, we consider the (weak) drop property, weak property (a), and property (w) for closed convex sets. Here we give some relations between those properties. Particularly, we prove that C has (weak) property (a) if and only if the subdifferential mapping of Cº is (n-n) (respectively, (n-w)) upper semicontinuous and (weak) compact valued. This gives an extension of a theorem of Giles and the first author.

On some geometric properties of certain Köthe sequence spaces

Yunan Cui, Henryk Hudzik, Tao Zhang (1999)

Mathematica Bohemica

It is proved that if a Kothe sequence space X is monotone complete and has the weakly convergent sequence coefficient WCS ( X ) > 1 , then X is order continuous. It is shown that a weakly sequentially complete Kothe sequence space X is compactly locally uniformly rotund if and only if the norm in X is equi-absolutely continuous. The dual of the product space ( ⨁ i = 1 ∞ X i ) Φ of a sequence of Banach spaces ( X i ) i = 1 ∞ , which is built by using an Orlicz function Φ satisfying the Δ 2 -condition, is computed isometrically (i.e. the exact...

On some properties for dual spaces of Musielak-Orlicz function spaces

Zenon Zbąszyniak (2011)

Banach Center Publications

We will present relationships between the modular ρ* and the norm in the dual spaces ( L Φ ) * in the case when a Musielak-Orlicz space L Φ is equipped with the Orlicz norm. Moreover, criteria for extreme points of the unit sphere of the dual space ( L ⁰ Φ ) * will be presented.

On spreading c 0 -sequences in Banach spaces

Vassiliki Farmaki (1999)

Studia Mathematica

We introduce and study the spreading-(s) and the spreading-(u) property of a Banach space and their relations. A space has the spreading-(s) property if every normalized weakly null sequence has a subsequence with a spreading model equivalent to the usual basis of c 0 ; while it has the spreading-(u) property if every weak Cauchy and non-weakly convergent sequence has a convex block subsequence with a spreading model equivalent to the summing basis of c 0 . The main results proved are the following: (a)...

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