Some new characterizations and \(g\)-minimality and stability of \(g\)-bases in the Hilbert spaces (Q352563)

Summary: The concept of \(g\)-basis in the Hilbert spaces has been introduced by \textit{X.-X. Guo} [Abstr. Appl. Anal. 2012, Article ID 923729 (2012; Zbl 1266.46018)], generalizing the concept of the Schauder basis in Hilbert spaces. A~\(g\)-basis plays a similar role in \(g\)-frame theory as the Schauder basis plays in frame theory. In the present paper, we establish some important properties of \(g\)-bases in Hilbert spaces. In particular, we obtain a simple condition under which some important properties established by Guo [loc.\,cit.]\ remain true. With these conditions, we also establish some new interesting properties of \(g\)-bases which are related to \(g\)-minimality. Finally, we obtain a perturbation result about \(g\)-bases.