Fundamental aspects of vector-valued Banach limits (Q2817535)

Let \(X\) be a real Banach space. In the paper [Funct. Anal. Appl. 47, No. 4, 315--318 (2013); translation from Funkts. Anal. Prilozh. 47, No. 4, 82--86 (2013; Zbl 1326.40005)], \textit{R. Armario} and the authors extended the concept of Banach limits from bounded real-valued sequences to the vector-valued case as follows: a Banach limit on \(X\) is a bounded linear operator \(T:\ell^{\infty}(X) \rightarrow X\) of norm one which is shift-invariant and satisfies \(T((x_n)_{n\in \mathbb{N}})=\lim_{n\to \infty}x_n\) for all norm-convergent sequences \((x_n)_{n\in \mathbb{N}}\) in \(X\).NEWLINENEWLINE It was proved in [loc.\,cit.]\ that a Banach limit on \(X\) exists if \(X\) is 1-complemented in its bidual, and also that there is no Banach limit on \(c_0\).NEWLINENEWLINE In this paper, the authors show that, conversely, if \(X\) has a monotone Schauder basis and there exists a Banach limit on \(X\), then \(X\) must be 1-complemented in its bidual. They further prove that if there is a Banach limit on the space \(X_i\) for each \(i\in \mathbb{N}\), then there is also a Banach limit on the \(\ell^{\infty}\)-sum of \((X_i)_{i\in \mathbb{N}}\) (Section 2).NEWLINENEWLINE In the third section, the authors study variants of Lorentz's characterisation of almost convergence in the vector-valued setting.NEWLINENEWLINE In Section 4, the following result is proved: for any real Banach space \(X\), the set of all operators of the form \(T\circ \varphi\), where \(T:X \rightarrow X\) is an isometric isomorphism and \(\varphi\) is a Banach limit on \(X\), is a proper subset of the unit sphere of \(N_X\) (the space of all shift-invariant bounded linear operators from \(\ell^{\infty}(X)\) to \(X\)).NEWLINENEWLINE In the last section it is shown that the set of all Banach limits on \(X\) is a face of the unit ball of \(N_X\), provided that \(X\) has the Krein-Milman property.