Structural properties of weak cotype 2 spaces

A Banach space with a symmetric basis which is of weak cotype 2 but not of cotype 2

We prove that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2

Weak type $(2,H)$ and weak cotype $(2,H)$ of operator spaces

Recently an operator space version of type and cotype, namely type $(p,H)$ and cotype $(q,H)$ of operator spaces for $1\leq p \leq 2\leq q \leq \infty$ and a subquadratic and homogeneous Hilbetian operator space $H$ were introduced and investigated by the author. In this paper we define weak type $(2,H)$ (resp. weak cotype $(2,H)$) of operator spaces, which lies strictly between type $(2,H)$ (resp. cotype $(2,H)$) and type $(p,H)$ for all $1\leq p <2$ (resp. cotype $(q,H)$ for all $2<q \leq \infty$). This is an analogue of weak type 2 and weak cotype 2 in the Banach space case, so we develop a…

Cotype and summing properties in Banach spaces

It is well known in Banach space theory that for a finite dimensional space $E$ there exists a constant $c_E$, such that for all sequences $(x_k)_k \subset E$ one has \[ \summ_k \noo x_k \rrm \kl c_E \pl \sup_{\eps_k \pm 1} \noo \summ_k \eps_k x_k \rrm \pl .\] Moreover, if $E$ is of dimension $n$ the constant $c_E$ ranges between $\sqrt{n}$ and $n$. This implies that absolute convergence and unconditional convergence only coincide in finite dimensional spaces. We will characterize Banach spaces $X$, where the constant $c_E \sim \sqrt{n}$ for all finite dimensional subspaces. More generally, we…

Constructions of Majorizing Measures, Bernoulli processes and cotype

We present three methods to construct majorizing measures in various settings. These methods are based on direct constructions of increasing sequences of partitions through a simple exhaustion procedure rather than on the construction of well separated ultrametric subspaces. The first scheme of construction provides a simple unified proof of the Majorizing Measure Theorem for Gaussian processes and of the following fact. If $A,B$ are balanced convex sets in a vector space, and if $A$ is sufficiently convex, a control of the covering numbers $N(A,\varepsilon B)$ for all $\varepsilon>0$ implies …

Abscissas of weak convergence of vector valued Dirichlet series

The abscissas of convergence, uniform convergence and absolute convergence of vector valued Dirichlet series with respect to the original topology and with respect to the weak topology $σ(X,X')$ of a locally convex space $X$, in particular of a Banach space $X$, are compared. The relation of their coincidence with geometric or topological properties of the underlying space $X$ is investigated. Cotype in the context of Banach spaces, and nuclearity and certain topological invariants for Fréchet spaces play a relevant role.

论巴拿赫空间之间粗映射的扩张性与几何保持性

On the expansiveness of coarse maps between Banach spaces and geometry preservation

该研究提出巴拿赫空间间新的可嵌入性概念，证明其严格弱于粗可嵌入性，借助度量余型技术研究余型保持与ℓ_p空间嵌入，结合Kalton非线性不变量对比，还探讨了ℓ_∞嵌入问题。