Related papers: A generalised mid summability in Banach spaces

We prove when a Banach ideal of linear operators defined, or characterized, by the transformation of vector-valued sequences is maximal. Known results are recovered as particular cases and new information is obtained. To accomplish this…

Given any Banach space $X$, let $L_2^X$ denote the Banach space of all measurable functions $f:[0,1]\to X$ for which ||f||_2:=(int_0^1 ||f(t)||^2 dt)^{1/2} is finite. We show that $X$ is a UMD--space (see \cite{BUR:1986}) if and only if…

Banach spaces that are complemented in the second dual are characterised precisely as those spaces $X$ which enjoy the property that for every amenable semigroup $S$ there exists an $X$-valued analogue of an invariant mean defined on the…

We define a monad M on a category of measurable bornological sets, and we show how this monad gives rise to a theory of vector-valued integration that is related to the notion of Pettis integral. We show that an algebra X of this monad is a…

Generalizing classical results of the theory of absolutely summing operators, in this paper we characterize the duals of a quite large class of Banach operator ideals defined or characterized by the transformation of vector-valued…

We introduce kernel-summability methods in Banach spaces using the vector-valued integrals and prove an analogue of the Silverman-Toeplitz Theorem for regular kernel-summability methods. We also show that if $X$ is a Banach space and one…

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…

A Banach space $X$ has the $2$-summing property if the norm of every linear operator from $X$ to a Hilbert space is equal to the $2$-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent…

In this paper, we deal with a notion of Banach space-valued mappings defined on a set consisting of finite graphs with uniformly bounded vertex degree. These functions will be endowed with certain boundedness and additivity criteria. We…

We formalize the observation that the same summability methods converge in a Banach space $X$ and its dual $X^*$. At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on $X$ and $X^*$…

We extend the $\lambda$-theory of operator spaces given by Defant and Wiesner (2014), that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach $*$-algebras. Given…

The sequence space of all real-valued sequences, denoted $Seq(\mathbb{R})$, is typically investigated through the lens of infinite-dimensional vector spaces, utilizing Banach space norms or Schauder bases. This work proposes a…

Let $X$ be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces $E$ of $X$-valued sequences, the sets $E-\bigcup _{q\in\Gamma}\ell_{q}(X)$, where $\Gamma$ is any subset of $(0,\infty]$, and $E-c_{0}(X)$ contain…

In this paper we define and study a vector-valued sequence space, called the space of anisotropic $(s,q,r)$-summable sequences, that generalizes the classical space of $(s; q)$-mixed sequences (or mixed $(s; q)$-summable sequences).…

Our principal result is the following. Let $X$ and $Y$ be Banach spaces, let $G$ be a locally compact abelian group, and let $K$ be an operator valued kernel defined on $G$ with values in the space of bounded linear operators from $X$ to…

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…

Let $E_{1},...,E_{m},F$ be Banach spaces. The index of summability of $\left(E_{1}\times\cdots\times E_{m},F\right) $ is a kind of measure of how far the $m$-linear operators $T:E_{1}\times\cdots\times E_{m}\rightarrow F$ are from being…

This work performs a study of the category of complete matrix-normed spaces, called matricial Banach spaces. Many of the usual constructions of Banach spaces extend in a natural way to matricial Banach spaces, including products, direct…

Let $Z$ and $X$ be Banach spaces. Suppose that $X$ is Asplund. Let $\mathcal{M}$ be a bounded set of operators from $Z$ to $X$ with the following property: a bounded sequence $(z_n)_{n\in \mathbb{N}}$ in $Z$ is weakly null if, for each $M…

A banach space X is a normed vector space, which is complete with respect to the metric induced by the norm. Given a bounded linear operator T acting on a banach space X, T is said to attain its norm if there is a unit vector z in X, such…