\(g\)-frames and modular Riesz bases in Hilbert \(C^\ast\)-modules (Q2911900)

The notion of \(g\)-frame in the setting of Hilbert spaces was defined by \textit{W. Sun} [J. Math. Anal. Appl. 322, No.~1, 437--452 (2006; Zbl 1129.42017)] and the concept of operator-valued frame was defined by \textit{V. Kaftal}, \textit{D. R. Larson} and \textit{S. Zhang} [Trans. Am. Math. Soc. 361, No.~12, 6349--6385 (2009; Zbl 1185.42032)]. The authors of the present paper introduce modular Riesz bases and modular \(g\)-Riesz bases in Hilbert \(C^\ast\)-modules and investigate their analogous properties to the Riesz bases and \(g\)-Riesz bases in the framework of Hilbert spaces. Using the fact that every finitely or countably generated Hilbert \(C^\ast\)-module over a unital \(C^\ast\)-algebra has a standard Parseval frame, they characterize \(g\)-frames, modular Riesz bases and modular \(g\)-Riesz bases.