关于宽$(s)$序列及其对某些算子类的应用
On wide-$(s)$ sequences and their applications to certain classes of operators
摘要：
Banach空间中的一个基本序列称为宽- $(s)$，如果它是有界的并且支配求和基。（宽- $(s)$序列最初由I.~ Singer将其称为$P^*$ -序列）。这些序列和它们的量化版本，称为$λ$ -宽- $(s)$序列，被用来表征各种类别的运营商之间的巴拿赫空间，如弱紧，Tauberian，超Tauberian运营商，以及一个新的中间类介绍这里，强Tauberian运营商。这是一个nonlocalizable类，但形成一个开放的半群，并关闭下的自然操作，如采取双伴随。例如，证明了一个算子是非弱紧的当且仅当对每个$\varepsilon >0$，它将某个$(1+\varepsilon)$ -宽- $(s)$ -序列映射到一个宽- $(s)$-序列.这产生了定量的三角形阵列结果，其表征了由于R.C.~ James.证明了一个算子是非Tauberian的（分别是非强Tauberian）当且仅当对于每个$\varepsilon>0$，它将某个$(1+\varepsilon)$ -宽- $(s)$序列映射为一个范数-收敛序列（分别为其图像具有小于$\varepsilon$的直径的序列）。这被应用于获得超Tauberian算子的直接"有限“特征，以及以下特征，其加强了M.~ González和A.~ Mart{\'s}nez-Abejón：一个算子是非超陶伯算子当且仅当对每个$\varepsilon>0$，存在任意长度的有限$(1+\varepsilon)$ -wide- $(s)$序列，其像至多有范数$\varepsilon$ .
英文摘要：
A basic sequence in a Banach space is called wide-$(s)$ if it is bounded and dominates the summing basis. (Wide-$(s)$ sequences were originally introduced by I.~Singer, who termed them $P^*$-sequences). These sequences and their quantified versions, termed $λ$-wide-$(s)$ sequences, are used to characterize various classes of operators between Banach spaces, such as the weakly compact, Tauberian, and super-Tauberian operators, as well as a new intermediate class introduced here, the strongly Tauberian operators. This is a nonlocalizable class which nevertheless forms an open semigroup and is closed under natural operations such as taking double adjoints. It is proved for example that an operator is non-weakly compact iff for every $\varepsilon >0$, it maps some $(1+\varepsilon)$-wide-$(s)$-sequence to a wide-$(s)$ sequence. This yields the quantitative triangular arrays result characterizing reflexivity, due to R.C.~James. It is shown that an operator is non-Tauberian (resp. non-strongly Tauberian) iff for every $\varepsilon>0$, it maps some $(1+\varepsilon)$-wide-$(s)$ sequence into a norm-convergent sequence (resp. a sequence whose image has diameter less than $\varepsilon$). This is applied to obtain a direct ``finite'' characterization of super-Tauberian operators, as well as the following characterization, which strengthens a recent result of M.~González and A.~Mart{\'ı}nez-Abejón: An operator is non-super-Tauberian iff there are for every $\varepsilon>0$, finite $(1+\varepsilon)$-wide-$(s)$ sequences of arbitrary length whose images have norm at most $\varepsilon$.
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