New inequalities for \(g\)-frames in Hilbert \(C^\ast\)-modules (Q2830107)

The notion of \(g\)-frames in Hilbert \(C^*\)-modules was introduced by \textit{A. Khosravi} and \textit{B. Khosravi} [Int. J. Wavelets Multiresolut. Inf. Process. 6, No. 3, 433--446 (2008; Zbl 1153.46035)] and their inequalities were investigated by \textit{X.-C. Xiao} and \textit{X.-M. Zeng} [J. Math. Anal. Appl. 363, No. 2, 399--408 (2010; Zbl 1189.46050)]. The author obtains new equalities and inequalities for \(g\)-frames in Hilbert \(C^*\)-modules. First, by using techniques of operator theory in Hilbert \(C^*\)-module, he presents new equalities and inequalities for \(g\)-frames in Hilbert \(C^*\)-modules. Then, he gives another equality and inequality with the Moore-Penrose inverse of adjointable operators in Hilbert \(C^*\)-modules and shows that they generalize Theorems 4.1 and 4.2 in [Xiao and Zeng, loc. cit.].