Arithmetic separation and Banach-Saks sets (Q442503)

A bounded non-empty subset \(D\) of a Banach space \(X\) is called a Banach-Saks set if every sequence in \(D\) has a subsequence \((x_n)_{n\in \mathbb{N}}\) such that the sequence of Cesàro means \((m^{-1}\sum_{n=1}^m x_n)_{m\in \mathbb{N}}\) is norm-convergent in \(X\).NEWLINENEWLINE The arithmetic separation of a bounded sequence \((x_n)_{n\in \mathbb{N}}\) in \(X\) is defined by NEWLINE\[NEWLINE \text{asep}(x_n):=\inf\left\|\frac{1}{m}\left(\sum_{n\in A}x_n-\sum_{n\in B}x_n\right) \right\|,NEWLINE\]NEWLINE NEWLINEwhere the infimum is taken over all \(m\in \mathbb{N}\) and all subsets \(A,B\subseteq \mathbb{N}\) with \(|{A}|=| {B}|=m\) and \(\max A<\min B\).NEWLINENEWLINE The author first shows that \(D\) is a Banach-Saks set if and only if \(\text{asep}(x_n)=0\) for every sequence \((x_n)_{n\in \mathbb{N}}\) in \(D\) and then introduces the measure NEWLINE\[NEWLINE \varphi(D)=\sup \{\text{asep} (x_n):(x_n)\subseteq D\} NEWLINE\]NEWLINE to quantify how ``far \(D\) is from being a Banach-Saks set''.NEWLINENEWLINE The measure \(\varphi\) is related to the measures \(\beta\) and \(\gamma\) of noncompactness respectively weak noncompactness, which use the separation respectively convex separation of a sequence instead of the arithmetic separation in their definition, by the inequalities NEWLINE\[NEWLINE \gamma(D)\leq\varphi(D)\leq\beta(D). NEWLINE\]NEWLINE A similar measure for the so called alternate signs Banach-Saks property is also introduced and their basic properties are established. Among other results it is shown that, if \(X\) does not have the Banach-Saks property, i.\,e., the closed unit ball \(B_X\) is not a Banach-Saks set, then one already has \(\varphi(B_X)\geq 1\).NEWLINENEWLINE For the James spaces \(J_p\) with \(1<p<\infty\) it is proved that \(\gamma(D)=\varphi(D)\) for every bounded subset \(D\subseteq J_p\).NEWLINENEWLINE In the last section of the paper, some applications of the Banach-Saks and alternate Banach-Saks measure to the polygon interpolation method for operators on Banach \(N\)-tuples are given.