A separable Banach space X contains ${\ell }_1$ isomorphically if and only if X has a bounded fundamental total $w{c}_0*$-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total $w{c}_0*$-biorthogonal system.

A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C\left({\omega }^{\omega }\right)$ then X contains a copy of c₀. Moreover,...

We study universal Dirichlet series with respect to overconvergence, which are absolutely convergent in the right half of the complex plane. In particular we obtain estimates on the growth of their coefficients. We can then compare several classes of universal Dirichlet series.

We introduce higher order spreading models associated to a Banach space X. Their definition is based on ℱ-sequences ${\left(x_s\right)}_{s\in ℱ}$ with ℱ a regular thin family and on plegma families. We show that the higher order spreading models of a Banach space X form an increasing transfinite hierarchy ${\left({ℳ}_{\xi }\left(X\right)\right)}_{\xi <\omega ₁}$. Each ${ℳ}_{\xi }\left(X\right)$ contains all spreading models generated by ℱ-sequences ${\left(x_s\right)}_{s\in ℱ}$ with order of ℱ equal to ξ. We also study the fundamental properties of this hierarchy.

Let us denote by C(α) the classical Banach space C(K) when K is the interval of ordinals [1,α] endowed with the order topology. In the present paper, we give an answer to a 1960 Bessaga and Pełczyński question by providing tight bounds for the Banach-Mazur distance between C(ω) and any other C(K) space which is isomorphic to it. More precisely, we obtain lower bounds L(n,k) and upper bounds U(n,k) on d(C(ω),C(ωⁿk)) such that U(n,k) - L(n,k) < 2 for all 1 ≤ n, k < ω.

Let X be an infinite dimensional separable Banach space. There exists a hypercyclic operator on X which is equal to the identity operator on an infinite dimensional closed subspace of X.

We study the (I)-envelopes of the unit balls of Banach spaces. We show, in particular, that any nonreflexive space can be renormed in such a way that the (I)-envelope of the unit ball is not the whole bidual unit ball. Further, we give a simpler proof of James' characterization of reflexivity in the nonseparable case. We also study the spaces in which the (I)-envelope of the unit ball adds nothing.

We give new characterizations of Banach spaces not containing ${\ell }_1$ in terms of integral and $p$-dominated polynomials, extending to the polynomial setting a result of Cardassi and more recent results of Rosenthal.

We study conditions on an infinite dimensional separable Banach space $X$ implying that $X$ is the only non-trivial invariant subspace of $X^{**}$ under the action of the algebra $𝔸\left(X\right)$ of biconjugates of bounded operators on $X$: $𝔸\left(X\right)=\{T^{**}T\in ℬ\left(X\right)\}$. Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of $c_0$, and show in particular that any space which does not contain ${\ell }_1$ and has property (u) of Pelczynski is simple.

Let $X$ and $E$ be a Banach space and a real Banach lattice, respectively, and let $\Gamma$ denote an infinite set. We give concise proofs of the following results: (1) The dual space $X^{*}$ contains an isometric copy of $c_0$ iff $X^{*}$ contains an isometric copy of ${\ell }_{\infty }$, and (2) $E^{*}$ contains a lattice-isometric copy of $c_0\left(\Gamma \right)$ iff $E^{*}$ contains a lattice-isometric copy of ${\ell }_{\infty }\left(\Gamma \right)$.

We introduce the definition of $p$-limited completely continuous operators, $1\le p<\infty$. The question of whether a space of operators has the property that every $p$-limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$-limited completely continuous evaluation operators.

It is shown that the weak $L^p$ spaces ${\ell }^{p,\infty },L^{p,\infty }[0,1]$, and $L^{p,\infty }[0,\infty )$ are isomorphic as Banach spaces.