An extension of Schreier unconditionality (Q931395)

A normalized basic sequence \((x_i)_{i=1}^\infty\) in a Banach space is called Schreier unconditional if, for some \(C<\infty\), all scalars \((a_i)_{i=1}^\infty\) and all Schreier sets \(F\subseteq\mathbb N\) (i.e., \(\#\,F\leq \min F\)), \[ \| \sum_{i\in F} a_i x_i\| \leq C\| \sum_{i=1}^\infty a_i x_i\| \;. \] This paper extends the result that every normalized weakly null sequence in a Banach space admits a \((2+\varepsilon)\)-Schreier unconditional subsequence to arrays, \((x_{i,j})_{(i,j)\in I}\), where \(I = \{(i,j) \in \mathbb N\times \mathbb N :i\leq j\}\). The rows, \((x_{i,j})_{j\geq i}\), are assumed to be seminormalized weakly null sequences. It is proved that there is a regular subarray \((y_{i,j})_I\) satisfying \[ \| \sum_{j\in F} a_{k_0,j} y_{k_0,j}\| \leq (2+\varepsilon)\,\|\sum_{(i,j)\in I} a_{i,j} y_{i,j}\| \] for all scalars \((a_{i,j})_I\), \(k_0\in\mathbb N\) and Schreier sets \(F\subseteq\mathbb N\) with \(k_0\leq\min F\). A regular subarray is one which, roughly, consists of taking a subsequence of each row so that the new array respects the original reverse lexicographic ordering. In addition, a regular subarray is basic in this natural order \((y_{1,1}, y_{1,2}, y_{2,2}, y_{1,3},\dots)\). Applications are given, extending results of [\textit{G.\,Androulakis, E.\,Odell, Th.\,Schlumprecht, N.\,Tomczak--Jaegermann}, Can.\ J.\ Math.\ 57, No.\,~4, 673--707 (2005; Zbl 1090.46004)] and [\textit{Th.\,Schlumprecht}, Isr.\ J.\ Math.\ 76, 1--2, 81--95 (1991; Zbl 0796.46007)], to the problem of showing that certain Banach spaces \(X\) admit a subspace \(Y\) and an operator \(T\in {\mathcal L}(Y,X)\) which is not of the form \(\lambda I+K\), \(I\) being the inclusion operator from \(Y\) into \(X\) and \(K\) being a compact operator.