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...

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.

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = $\overline{f^{\circ }\overline{\varphi }}$ for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z 1, ..., z n) = $({\lambda }_1{z}_{i_1},...,{\lambda }_n{z}_{i_n})$ for |λ j| = 1, 1 ≤ j ≤ n, and (i 1; ..., i n)is some permutation of the integers from...

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...

We characterize Banach lattices under which each b-weakly compact (resp. b-AM-compact, strong type (B)) operator is L-weakly compact (resp. M-weakly compact).