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### 2-summing multiplication operators

Studia Mathematica

Let 1 ≤ p < ∞, $={\left(Xₙ\right)}_{n\in ℕ}$ be a sequence of Banach spaces and ${l}_{p}\left(\right)$ the coresponding vector valued sequence space. Let $={\left(Xₙ\right)}_{n\in ℕ}$, $={\left(Yₙ\right)}_{n\in ℕ}$ be two sequences of Banach spaces, $={\left(Vₙ\right)}_{n\in ℕ}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator ${M}_{}:{l}_{p}\left(\right)\to {l}_{q}\left(\right)$ by ${M}_{}\left({\left(xₙ\right)}_{n\in ℕ}\right):={\left(Vₙ\left(xₙ\right)\right)}_{n\in ℕ}$. We give necessary and sufficient conditions for ${M}_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞.

### A class of Banach sequence spaces analogous to the space of Popov

Czechoslovak Mathematical Journal

Hagler and the first named author introduced a class of hereditarily ${l}_{1}$ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily ${l}_{p}$ Banach spaces for $1\le p<\infty$. Here we use these spaces to introduce a new class of hereditarily ${l}_{p}\left({c}_{0}\right)$ Banach spaces analogous of the space of Popov. In particular, for $p=1$ the spaces are further examples of hereditarily ${l}_{1}$ Banach spaces failing the Schur property.

### A disjointness type property of conditional expectation operators

Colloquium Mathematicae

We give a characterization of conditional expectation operators through a disjointness type property similar to band-preserving operators. We say that the operator T:X→ X on a Banach lattice X is semi-band-preserving if and only if for all f, g ∈ X, f ⊥ Tg implies that Tf ⊥ Tg. We prove that when X is a purely atomic Banach lattice, then an operator T on X is a weighted conditional expectation operator if and only if T is semi-band-preserving.

### A Natural Class of Sequential Banach Spaces

Bulletin of the Polish Academy of Sciences. Mathematics

We introduce and study a natural class of variable exponent ${\ell }^{p}$ spaces, which generalizes the classical spaces ${\ell }^{p}$ and c₀. These spaces will typically not be rearrangement-invariant but instead they enjoy a good local control of some geometric properties. Some geometric examples are constructed by using these spaces.

### A note on continuous restrictions of linear maps between Banach spaces.

Acta Mathematica Universitatis Comenianae. New Series

### About the Banach envelope of l1,∞.

Revista Matemática Complutense

### Acción dual de las copias, isométricas asintóticamente, de lp (1 ≤ p &lt; ∞) y c0.

Collectanea Mathematica

### Algebraic and topological properties of some sets in ℓ₁

Colloquium Mathematicae

For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series ${\sum }_{n=1}^{\infty }x\left(n\right)$. Guthrie and Nymann proved that E(x) is one of the following types of sets: () a finite union of closed intervals; () homeomorphic to the Cantor set; homeomorphic to the set T of subsums of ${\sum }_{n=1}^{\infty }b\left(n\right)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable. We...

### Amenability for discrete convolution semigroup algebras.

Mathematica Scandinavica

### An amalgamation of the Banach spaces associated with James and Schreier, Part I: Banach-space structure

Banach Center Publications

We create a new family of Banach spaces, the James-Schreier spaces, by amalgamating two important classical Banach spaces: James' quasi-reflexive Banach space on the one hand and Schreier's Banach space giving a counterexample to the Banach-Saks property on the other. We then investigate the properties of these James-Schreier spaces, paying particular attention to how key properties of their 'ancestors' (that is, the James space and the Schreier space) are expressed in them. Our main results include...

### An amalgamation of the Banach spaces associated with James and Schreier, Part II: Banach-algebra structure

Banach Center Publications

The James-Schreier spaces, defined by amalgamating James' quasi-reflexive Banach spaces and Schreier space, can be equipped with a Banach-algebra structure. We answer some questions relating to their cohomology and ideal structure, and investigate the relations between them. In particular we show that the James-Schreier algebras are weakly amenable but not amenable, and relate these algebras to their multiplier algebras and biduals.

### Approximation problems in modular spaces of double sequences.

Publicacions Matemàtiques

Let X denote the space of all real, bounded double sequences, and let Φ, φ, Γ be φ-functions. Moreover, let Ψ be an increasing, continuous function for u ≥ 0 such that Ψ(0) = 0.In this paper we consider some spaces of double sequences provided with two-modular structure given by generalized variations and the translation operator (...).

### Asymptotic uniform moduli and Kottman constant of Orlicz sequence spaces.

Revista Matemática Complutense

### Banach-Saks property in some Banach sequence spaces

Annales Polonici Mathematici

It is proved that for any Banach space X property (β) defined by Rolewicz in  implies that both X and X* have the Banach-Saks property. Moreover, in Musielak-Orlicz sequence spaces, criteria for the Banach-Saks property, the near uniform convexity, the uniform Kadec-Klee property and property (H) are given.

### Bicontractive projections in sequence spaces and a few related kinds of maps

Commentationes Mathematicae Universitatis Carolinae

### Bounding Subsets of a Banach Space.

Mathematische Annalen

### Brèves communications. Comparaison de suites convergentes

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

### Can $ℬ\left({\ell }^{p}\right)$ ever be amenable?

Studia Mathematica

It is known that $ℬ\left({\ell }^{p}\right)$ is not amenable for p = 1,2,∞, but whether or not $ℬ\left({\ell }^{p}\right)$ is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if $ℬ\left({\ell }^{p}\right)$ is amenable for p ∈ (1,∞), then so are ${\ell }^{\infty }\left(ℬ\left({\ell }^{p}\right)\right)$ and ${\ell }^{\infty }\left(\left({\ell }^{p}\right)\right)$. Moreover, if ${\ell }^{\infty }\left(\left({\ell }^{p}\right)\right)$ is amenable so is ${\ell }^{\infty }\left(,\left(E\right)\right)$ for any index set and for any infinite-dimensional ${ℒ}^{p}$-space E; in particular, if ${\ell }^{\infty }\left(\left({\ell }^{p}\right)\right)$ is amenable for p ∈ (1,∞), then so is ${\ell }^{\infty }\left(\left({\ell }^{p}\oplus \ell ²\right)\right)$. We show that ${\ell }^{\infty }\left(\left({\ell }^{p}\oplus \ell ²\right)\right)$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter over ℕ, we exhibit...

### Cantor-Bernstein theorems for Orlicz sequence spaces

Banach Center Publications

For two Banach spaces X and Y, we write $di{m}_{\ell }\left(X\right)=di{m}_{\ell }\left(Y\right)$ if X embeds into Y and vice versa; then we say that X and Y have the same linear dimension. In this paper, we consider classes of Banach spaces with symmetric bases. We say that such a class ℱ has the Cantor-Bernstein property if for every X,Y ∈ ℱ the condition $di{m}_{\ell }\left(X\right)=di{m}_{\ell }\left(Y\right)$ implies the respective bases (of X and Y) are equivalent, and hence the spaces X and Y are isomorphic. We prove (Theorems 3.1, 3.3, 3.5) that the class of Orlicz sequence spaces generated by regularly...

### Characterizations of completeness of normed spaces through weakly unconditionally Cauchy series

Czechoslovak Mathematical Journal

In this paper we obtain two new characterizations of completeness of a normed space through the behaviour of its weakly unconditionally Cauchy series. We also prove that barrelledness of a normed space $X$ can be characterized through the behaviour of its weakly-$*$ unconditionally Cauchy series in ${X}^{*}$.

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