Convex unconditionality and summability of weakly null sequences

Almost isometric constants for partial unconditionality

We discuss optimal constants of certain projections on subsequences of weakly null sequences. Positive results yield constants arbitrarily close to $1$ for Schreier type projections, and arbitrarily close to $1$ for Elton type projections under the assumption that the weakly null sequence admits no subsequence generating a $c_0$ spreading model. As an application, we prove that a weakly null sequence admitting a spreading model not equivalent to the $c_0$ basis has a quasi-greedy subsequence with quasi-greedy constant arbitrarily close to $1$.

Partial Unconditionality

J. Elton proved that every normalized weakly null sequence in a Banach space admits a subsequence that is nearly unconditional which is a weak form of unconditionality. The notion of near-unconditionality is quantified by a constant $K(δ)$ depending on a parameter $δ\in (0,1]$. It is unknown if $\sup_{δ>0} K(δ) < \infty$. This problem turns out to be closely related to the question whether every infinite-dimensional Banach space contains a quasi-greedy basic sequence. The notion of a quasi-greedy basic sequence was introduced recently by S. V. Konyagin and V. N. Temlyakov. We present an extens…

关于 Banach 空间中可和性的一个注记

A note on summability in Banach spaces

本文研究 Banach 空间中算子级数的绝对可和性，在 Asplund 空间与弱无条件 Cauchy 等条件下证明 Dunford-Pettis 算子对应级数绝对收敛，并用于重证向量测度有限变差结果。