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.

It is proved that a separable Banach space X admits a representation $X=X_1+X_2$ as a sum (not necessarily direct) of two infinite-codimensional closed subspaces $X_1$ and $X_2$ if and only if it admits a representation $X=A_1\left(Y_1\right)+A_2\left(Y_2\right)$ as a sum (not necessarily direct) of two infinite-codimensional operator ranges. Suppose that a separable Banach space X admits a representation as above. Then it admits a representation $X=T_1\left(Z_1\right)+T_2\left(Z_2\right)$ such that neither of the operator ranges $T_1\left(Z_1\right)$, $T_2\left(Z_2\right)$ contains an infinite-dimensional closed subspace if and only...