We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space $K_{2n}$ such that $C\left(K_{2n}\right)$ has no uncountable (semi)biorthogonal sequence ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of at most 2n-1 points of $K_{2n}$, but has biorthogonal systems ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves that its is...

We introduce a new “weak” BMO-regularity condition for couples (X,Y) of lattices of measurable functions on the circle (Definition 3, Section 9), describe it in terms of the lattice $X^{1/2}{\left(Y^{\text{'}}\right)}^{1/2}$, and prove that this condition still ensures “good” interpolation for the couple $(X_A,Y_A)$ of the Hardy-type spaces corresponding to X and Y (Theorem 1, Section 9). Also, we present a neat version of Pisier’s approach to interpolation of Hardy-type subspaces (Theorem 2, Section 13). These two main results of the paper are...

This survey features some recent developments concerning the bounded approximation property in Banach spaces. As a central theme, we discuss the weak bounded approximation property and the approximation property which is bounded for a Banach operator ideal. We also include an overview around the related long-standing open problem: Is the approximation property of a dual Banach space always metric?

One of the most important methods used in literature to introduce new properties in a Banach space E, consists in establishing some non trivial relationships between different classes of subsets of E. For instance, E is reflexive, or has finite dimension, if and only if every bounded subset is weakly relatively compact or norm relatively compact, respectively.On the other hand, Banach spaces of the type C(K) and Lp(μ) play a vital role in the general theory of Banach spaces. Their structure is so...

Criteria for compactly locally uniformly rotund points in Orlicz spaces are given.

We show that the Hilbert space is coarsely embeddable into any ${\ell }_p$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into ${\ell }_p$ are equivalent for 1 ≤ p < 2.

Let X be a complex Banach space and let Bloch(X) denote the space of X-valued analytic functions on the unit disc such that $su{p}_{\left|z\right|<1}(1-|z\left|²\right)\left|\right|f^{\text{'}}\left(z\right)\left|\right|<\infty$. A sequence (Tₙ)ₙ of bounded operators between two Banach spaces X and Y is said to be an operator-valued multiplier between Bloch(X) and ℓ₁(Y) if the map ${\sum }_{n=0}^{\infty }xₙzⁿ\to \left(Tₙ\left(xₙ\right)\right)ₙ$ defines a bounded linear operator from Bloch(X) into ℓ₁(Y). It is shown that if X is a Hilbert space then (Tₙ)ₙ is a multiplier from Bloch(X) into ℓ₁(Y) if and only if $su{p}_k{\sum }_{n=2^k}^{2^{k+1}}\left|\right|Tₙ\left|\right|²<\infty$. Several results about Taylor coefficients of vector-valued...

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...