R. Deville and J. Rodríguez proved that, for every Hilbert generated space $X$, every Pettis integrable function $f:[0,1]\to X$ is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space $X$ and a scalarly null (hence Pettis integrable) function from $[0,1]$ into $X$, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from $[0,1]$ (mostly) into $C\left(K\right)$ spaces. We focus in more detail on the behavior...

We show that as soon as $c_0$ embeds complementably into the space of all weakly compact operators from $X$ to $Y$, then it must live either in $X^{*}$ or in $Y$.

Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $\otimes {̂}_{\epsilon }^{n}C\left(K\right)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $\otimes {̂}_{\epsilon }^{n}C\left(K\right)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X\otimes {̂}_{\epsilon }Y$ contains a complemented copy of c₀ if one of the infinite-dimensional Banach...

A necessary and sufficient condition is given for a rearrangement invariant function space to contain a complemented isomorphic copy of l1(l2).

We study geodesic completeness for left-invariant Lorentz metrics on solvable Lie groups.

Let ${[X_0,X_1]}_t$, 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both $X_0$ and $X_1$ will act boundedly on each ${[X_0,X_1]}_t$. Let $T_t$ denote such an operator when considered on ${[X_0,X_1]}_t$, and $\sigma \left(T_t\right)$ denote its spectrum. We are motivated by the question of whether or not the map $t\to \sigma \left(T_t\right)$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: $t\to {\left(\sigma \left(T_t\right)\right)}^{\wedge }$ (polynomially convex hull) and $t\to {\partial }_e\left(\sigma \left(T_t\right)\right)$ (boundary of the polynomially convex...

We prove a conjecture of Wojtaszczyk that for 1 ≤ p < ∞, p ≠ 2, $H_p\left(\right)$ does not admit any norm one projections with dimension of the range finite and greater than 1. This implies in particular that for 1 ≤ p < ∞, p ≠ 2, $H_p$ does not admit a Schauder basis with constant one.

In this note we study some properties concerning certain copies of the classic Banach space $c_0$ in the Banach space $ℒ\left(X,Y\right)$ of all bounded linear operators between a normed space $X$ and a Banach space $Y$ equipped with the norm of the uniform convergence of operators.