On stability of G-frames in Hilbert pro-\(C^\ast\)-modules
The notion of a frame in a Hilbert space was first introduced by \textit{R. J. Duffin} and \textit{A. C. Schaffer} in [Trans. Am. Math. Soc. 72, 341--366 (1952; Zbl 0049.32401)]. Since then, the concept has seen many generalizations and variations, notably the notion of a g-frame in a Hilbert space due to [\textit{W. Sun}, J. Math. Anal. Appl. 322, No. 1, 437--452 (2006; Zbl 1129.42017)]. In a different direction, replacing the field of scalars in the definition of a complex Hilbert space by an arbitrary \(C^\ast\)-algebra leads to the notion of a Hilbert \(C^\ast\)-module. Frames were extended to this setting in [\textit{M. Frank} and \textit{D. R. Larson}, J. Oper. Theory 48, No. 2, 273--314 (2002; Zbl 1029.46087)]. In the present paper, g-frames in Hilbert pro-\(C^\ast\)-modules are studied. Specifically, a perturbation result for ordinary frames in Hilbert spaces due to \textit{P. G. Casazza} and \textit{O. Christensen} [J. Fourier Anal. Appl. 3, No. 5, 543--557 (1997; Zbl 0895.47007)] is extended to this setting. We mention that g-frames in Hilbert \(C^\ast\)-modules were first studied in [\textit{A. Khosravi} and \textit{B. Khosravi}, Int. J. Wavelets Multiresolut. Inf. Process. 6, No. 3, 433--446 (2008; Zbl 1153.46035)]. See also [\textit{N. C. Phillips}, J. Oper. Theory 19, No. 1, 159--195 (1988; Zbl 0662.46063)] for a reference on Hilbert pro-\(C^\ast\)-modules.
scientific article; zbMATH DE number 7342730
scientific article