Related papers: Absolutely summing linear operators into spaces wi…

Unifying several directions of the development of the study of summing multilinear operators between Banach spaces, we construct a general framework that studies, under one single definition, multilinear operators that are summing with…

Let $X$, $Y$ and $Z$ be Banach spaces and let $U$ be a subspace of $\mathcal{L}(X^*,Y)$, the Banach space of all operators from $X^*$ to $Y$. An operator $S: U \to Z$ is said to be $(\ell^s_p,\ell_p)$-summing (where $1\leq p <\infty$) if…

We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on a Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension…

Let $X,Y$ and $Z$ be Banach spaces, and let $\prod_p(Y,Z) (1\leq p<\infty)$ denote the space of $p$-summing operators from $Y$ to $Z$. We show that, if $X$ is a {\it \$}$_\infty$-space, then a bounded linear operator $T: X\hat…

In this paper, we develop the theory of absolutely summing multipolynomials. Among other results, we generalize and unify previous works of G. Botelho and D. Pellegrino concerning absolutely summing polynomials/multilinear mappings in…

Let $E$ be an operator space in the sense of the theory recently developed by Blecher-Paulsen and Effros-Ruan. We introduce a notion of $E$-valued non commutative $L_p$-space for $1 \leq p < \infty$ and we prove that the resulting operator…

Kwapie\'{n}'s theorem asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{p}$ is absolutely $\left( r,1\right) $-summing for $1/r=1-\left\vert 1/p-1/2\right\vert .$ When $p=2$ it recovers the famous Grothendieck's…

We prove that if q is in (1,\infty), Y is a Banach space and T is a linear operator defined on the space of finite linear combinations of (1,q)-atoms in R^n which is uniformly bounded on (1,q)-atoms, then T admits a unique continuous…

In this paper we prove coincidence results concerning spaces of absolutely summing multilinear mappings between Banach spaces. The nature of these results arises from two distinct approaches: the coincidence of two \textit{a priori}…

We provide quite sufficient conditions on the Banach spaces $E$ and $F$ in order to obtain the spaceability of the set of all linear operators from $E$ into $F$ which are $q$-compact but not $p$-compact. Also, under similar conditions over…

A famous result of S. Kwapie\'{n} asserts that a linear operator from a Banach space to a Hilbert space is absolutely $1$-summing whenever its adjoint is absolutely $q$-summing for some $1\leq q<\infty$; this result was recently extended to…

The objective of this study is to advance the theory concerning positive summing operators. Our focus lies in examining the space of positive strongly p-summable sequences and the space of positive unconditionally p-summable sequences. We…

In this paper we introduce an abstract approach to the notion of absolutely summing multilinear operators. We show that several previous results on different contexts (absolutely summing, almost summing, Cohen summing) are particular cases…

The aim of this paper is to start the study of multilinear generalizations of the classical ideals of linear operators of type $p$ and cotype $q$. As a first step in a theory we believe will be long and fruitful, we propose a notion of type…

Let $1 \leq p <\infty$. A sequence $\lef x_n \rig$ in a Banach space $X$ is defined to be $p$-operator summable if for each $\lef f_n \rig \in l^{w^*}_p(X^*)$, we have $\lef \lef f_n(x_k)\rig_k \rig_n \in l^s_p(l_p)$. Every norm…

Given a continuous $n$-homogeneous polynomial $P\colon E\longrightarrow F$ between Banach spaces and $1\leq q\leq p<\infty$, in this paper we investigate some properties concerning lineability and spaceability of the $(p;q)$-summing set of…

We give a new proof of a characterization of the closeness of the range of a continuous linear operator and of the closeness of the sum of two closed vector subspaces of a Banach space. Then we state sufficient conditions for the closeness…

Theorem: Let X and Y be two Banach spaces, Phi: X to Y a continuous, linear, surjective operator, and Psi: X to Y a completely continuous operator with bounded range. Then, one has dim{x in X : Phi(x)=Psi(x)} >= dim(Phi^{-1}(0)). Here dim…

We show that given a positive integer $m$, a real number $p\in\left[ 2,\infty\right)$ and $1\leq s<p^{\ast}$ the set of non--multiple $\left( r;s\right)$--summing $m$--linear forms on $\ell_{p}\times\cdots\times \ell_{p}$ contains, except…

We examine the chaotic behavior of certain continuous linear operators on infinite-dimensional Banach spaces, and provide several equivalent characterizations of when these operators have infinite topological entropy. For example, it is…