On vector-valued Banach limits
A linear functional \(\varphi: \ell^{\infty} \rightarrow \mathbb{R}\) is called a Banach limit if \(\varphi\) is positive (i.e., \(\varphi((x_n)_{n\in\mathbb{N}})\geq 0\) if \(x_n\geq 0\) for each \(n\in \mathbb{N}\)), invariant under the left-shift operator, and maps the constant sequence \((1,1,1,\dots)\) to \(1\). Every Banach limit is bounded with norm one and satisfies \(\varphi((x_n)_{n\in \mathbb{N}})=\lim_{n\to \infty}x_n\) for each convergent sequence \((x_n)_{n\in \mathbb{N}}\). The existence of Banach limits was already proved by S.\,Banach himself. In this paper, the authors give a generalisation of the concept of Banach limits to vector-valued sequences. They first note that a linear functional \(\varphi: \ell^{\infty} \rightarrow \mathbb{R}\) is a Banach limit if and only if \(\varphi\) is bounded with norm one, left-shift invariant, and satisfies \(\varphi((x_n)_{n\in \mathbb{N}})=\lim_{n\to \infty}x_n\) for every convergent sequence \((x_n)_{n\in \mathbb{N}}\). Thus they define, for a real Banach space \(X\), the set \(BL(X)\) of all vector-valued Banach limits on \(X\) as the set of all bounded linear operators \(T:\ell^{\infty}(X) \rightarrow X\) with \(\|T\|=1\) which are left-shift invariant and satisfy \(\varphi((x_n)_{n\in \mathbb{N}})=\lim_{n\to \infty}x_n\) for every norm-convergent sequence \((x_n)_{n\in \mathbb{N}}\) in \(X\). The authors show that for every real Banach space \(X\) one has \(BL(X^*)\neq \emptyset\), where \(X^*\) is the dual space of \(X\). On the other hand, for the space \(c_0\) of all real-valued null sequences (equipped with the supremum norm) one has \(BL(c_0)=\emptyset\). Also, if \(BL(X)\neq \emptyset\) and \(Y\) is a 1-complemented subspace of \(X\), then \(BL(Y)\neq \emptyset\). In [Acta Math., Uppsala 80, 167--190 (1948; Zbl 0031.29501)], \textit{G. G. Lorentz} defined a sequence \((x_n)_{n\in \mathbb{N}}\in \ell^{\infty}\) to be almost convergent to \(x\in \mathbb{R}\) if \(\varphi((x_n)_{n\in \mathbb{N}})=x\) holds for every Banach limit \(\varphi\). Lorentz proved in [loc.\,cit.]\ that \((x_n)_{n\in \mathbb{N}}\) is almost convergent to \(x\) if and only if \((x_n)_{n\in \mathbb{N}}\) is uniformly Cesàro convergent to \(x\), that is, \[ \lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^nx_{i+k}=x \;\text{uniformly in } k\in \mathbb{N}\cup\{0\}. \] The authors establish the following analogous result for the vector-valued case: if \((x_n)_{n\in \mathbb{N}}\) is a bounded sequence in \(X\), then \((x_n)_{n\in \mathbb{N}}\) is uniformly Cesàro convergent to \(0\) if and only if \(T((x_n)_{n\in \mathbb{N}})=0\) for every bounded linear operator \(T:\ell^{\infty}(X) \rightarrow X\) which is left-shift invariant.
scientific article; zbMATH DE number 6383397