Bessaga and Pełczyński showed that if $c_0$ embeds in the dual $X^{*}$ of a Banach space X, then ${\ell }^1$ embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of ${\ell }^1$ contains a copy of ${\ell }^1$ that is complemented in ${\ell }^1$. Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of $L^1[0,1]$ contains a copy of ${\ell }^1$ that is complemented in $L^1[0,1]$. In this note a traditional sliding hump argument is used to establish a simple mapping property of $c_0$ which simultaneously...

If $E=e_i$ and $F=f_i$ are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices $a_{i,j}$ with norm $\left|\right|a_{i,j}{\left|\right|}_{E\left(F\right)}=\left|\right|{\sum }_k\left|\right|{\sum }_l{a}_{k,l}{f}_l\left|\right|e_k\left|\right|$ embeds into L₁. This generalizes a recent result of Prochno and Schütt.

We say that a function from $X=C^L[0,1]$ is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Πₙ ⊂ X, where Πₙ denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general “shape” to preserve....

We consider the class of compact spaces $K_A$ which are modifications of the well known double arrow space. The space $K_A$ is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of $K_A$ spaces and on the isomorphic classification of the Banach spaces $C\left(K_A\right)$.

Given Banach spaces  $X$, $Y$ and a compact Hausdorff space  $K$, we use polymeasures to give necessary conditions for a multilinear operator from $C(K,X)$ into  $Y$ to be completely continuous (resp.  unconditionally converging). We deduce necessary and sufficient conditions for  $X$ to have the Schur property (resp.  to contain no copy of  $c_0$), and for  $K$ to be scattered. This extends results concerning linear operators.

We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_p$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_p$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T:l_{4/3}\times l_{4/3}\to l₂$ such that none of the associated linear operators is s-summing...

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set ${\Phi }^{-1}\left(E\right)$ is residual whenever E is residual in...