We establish Hölder-type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals.

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 study a higher-dimensional version of the standard notion of a gap formed by a finite sequence of ideals of the quotient algebra 𝓟(ω)/fin. We examine different types of such objects found in 𝓟(ω)/fin both from the combinatorial and the descriptive set-theoretic side.

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...

Let C(K) denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. Our main interest is in correlations between properties of the product of closed order intervals in C(K) and properties of the underlying space K. When K is finite, the product of two intervals in C(K) is always an interval....

Multiplication by harmonic representations of distributions, introduced by Li Banghe, is an extension of a certain product by radial (rotationally symmetric) mollifiers and therefore a strict extension of the Kami'{n}ski and Colombeau product.

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.

For a Banach space X, we show how the existence of a norm-one element u in X and a norm-one continuous bilinear mapping f: X x X --&gt; X satisfying f(x,u) = f(u,x) = x for all x in X, together with some more intrinsic conditions, can be utilized to characterize X as a member of some relevant subclass of the class of Banach spaces.