Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower-${\ell }_q$-tree estimate and let T be a bounded linear operator from X which satisfies an upper-${\ell }_p$-tree estimate. Then T factors through a subspace of ${\left(\sum Fₙ\right)}_{{\ell }_r}$, where (Fₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an $({\ell }_p,{\ell }_q)$ FDD. Similarly, let 1 < q < r < p < ∞ and let X be a separable reflexive Banach space satisfying an asymptotic lower-${\ell }_q$-tree...

Let 1 < p < ∞. Let X be a subspace of a space Z with a shrinking F.D.D. (Eₙ) which satisfies a block lower-p estimate. Then any bounded linear operator T from X which satisfies an upper-(C,p)-tree estimate factors through a subspace of ${\left(\sum Fₙ\right)}_{l_p}$, where (Fₙ) is a blocking of (Eₙ). In particular, we prove that an operator from $L_p$ (2 < p < ∞) satisfies an upper-(C,p)-tree estimate if and only if it factors through $l_p$. This gives an answer to a question of W. B. Johnson. We also prove that if X is...

We establish the existence of Banach spaces E and F isomorphic to complemented subspaces of each other but with $E^m\oplus Fⁿ$ isomorphic to $E^p\oplus F^q$, m, n, p, q ∈ ℕ, if and only if m = p and n = q.

We investigate isomorphic embeddings T: C(K) → C(L) between Banach spaces of continuous functions. We show that if such an embedding T is a positive operator then K is the image of L under an upper semicontinuous set-function having finite values. Moreover we show that K has a π-base of sets whose closures are continuous images of compact subspaces of L. Our results imply in particular that if C(K) can be positively embedded into C(L) then some topological properties of L, such as countable...

We introduce and study the spreading-(s) and the spreading-(u) property of a Banach space and their relations. A space has the spreading-(s) property if every normalized weakly null sequence has a subsequence with a spreading model equivalent to the usual basis of $c_0$; while it has the spreading-(u) property if every weak Cauchy and non-weakly convergent sequence has a convex block subsequence with a spreading model equivalent to the summing basis of $c_0$. The main results proved are the following: (a)...

This paper mainly concerns the topological nature of uniformly convexifiable sets in general Banach spaces: A sufficient and necessary condition for a bounded closed convex set C of a Banach space X to be uniformly convexifiable (i.e. there exists an equivalent norm on X which is uniformly convex on C) is that the set C is super-weakly compact, which is defined using a generalization of finite representability. The proofs use appropriate versions of classical theorems, such as James' finite tree...

We show that a Banach space X has an infinite dimensional reflexive subspace (quotient) if and only if there exist a Banach space Z and a non-isomorphic one-to-one (dense range) Tauberian (co-Tauberian) operator form X to Z (Z to X). We also give necessary and sufficient condition for the existence of a Tauberian operator from a separable Banach space to c0 which in turn generalizes a result of Johnson and Rosenthal. Another application of our result shows that if X** is separable, then there exists...

We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Y. Gordon (1970), we show that the constant $2$ appearing in the Amir-Cambern theorem may be replaced by $3$ for some class of subspaces. We achieve this by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces differs from the...

It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^{\xi }$ generated by subsequences $\left(e_{l_n}^{\xi }\right)$ and $\left(e_{m_n}^{\xi }\right)$, respectively, of the natural Schauder basis $\left(e_n^{\xi }\right)$ of $X^{\xi }$ are isomorphic if and only if $\left(e_{l_n}^{\xi }\right)$ and $\left(e_{m_n}^{\xi }\right)$ are equivalent. Further, $X^{\xi }$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $\left(e_n^{\xi }\right)$. It is also shown that there exists a complemented subspace spanned by a block basis of $\left(e_n^{\xi }\right)$, which is not isomorphic to a subspace generated by a subsequence of $\left(e_n^{\zeta }\right)$, for every $0\le \zeta \le \xi$....

On every subspace of $l_{\infty }\left(\mathbb{N}\right)$ which contains an uncountable $\omega$-independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of $l_{\infty }\left(\mathbb{N}\right)$ is infinite. This provides a partial answer to a question asked by Johnson and Odell.

We use one-dimensional differential inequalities to estimate the squareness and type of Banach spaces with modulus of convexity of power type two. The estimates obtained are sharp and the constants involved moderate.