A generalization of Gerisch's theorem on biorthogonal systems (Q2784962)

\textit{W. Gerisch} [Dokl. Akad. Nauk SSSR 200, 1018-1019 (1971; Zbl 0235.42008)] considered a biorthogonal pair of sequences \(\{ e_i\}\) and \(\{ g_i\}\) in a Hilbert space, each of them being complete, and a numerical sequence \(\{ \varepsilon_i=\pm 1\}\) such that the series \(\sum \varepsilon_i|(x,e_i)|^2\) and \(\sum \varepsilon_i|(x,g_i)|^2\) converge for every vector \(x\). It was proved that each of the above sequences of vectors is a basis (possibly conditional). NEWLINENEWLINENEWLINEThe author gives the following generalization: the biorthogonal sequences are considered in a dual pair of Banach spaces; instead of the convergence, the series are summable by certain linear methods, the basis property is replaced by an appropriate summability assertion. Some examples are given.