Farfar, R. Zarghami; Sadri, V.; Ahmadi, R. Some identities and inequalities for g-fusion frames. (English) Zbl 1457.42044 Probl. Anal. Issues Anal. 9(27), No. 2, 152-162 (2020). Summary: G-fusion frames, which are obtained from the combination of g-frames and fusion frames, were recently introduced for Hilbert spaces. In this paper, we present a new identity for g-frames, which was given by Najati for a special case. Also, by using the idea of this identity and the dual frames, some equalities and inequalities are presented for g-fusion frames. MSC: 42C15 General harmonic expansions, frames 46C99 Inner product spaces and their generalizations, Hilbert spaces 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) Keywords:g-frame; dual g-frame; g-fusion frame; dual g-fusion frame PDFBibTeX XMLCite \textit{R. Z. Farfar} et al., Probl. Anal. Issues Anal. 9(27), No. 2, 152--162 (2020; Zbl 1457.42044) Full Text: DOI arXiv MNR References: [1] Arabyani F., Minaei G. M., Anjidani E., “On Some Equalities and Inequalities for \(K\)-Frames”, Indian J. Pure. Appl. Math., 50:2 (2019), 297-308 · Zbl 1429.42033 [2] Balan R., Casazza P. G., Edidin D., Kutyniok G., “A New Identity for Parseval Frames”, Proc. Amer. Math. Soc., 135 (2007), 1007-1015 · Zbl 1136.42308 [3] Blocsli H., Hlawatsch H. F., Fichtinger H. G., “Frame-Theoretic analysis of oversampled filter bank”, IEEE Trans. Signal Processing, 46:12 (1998), 3256-3268 [4] Candes E. J., Donoho D. L., “New tight frames of curvelets and optimal representation of objects with piecewise \(C^2\) singularities”, Comm. Pure and App. Math., 57:2 (2004), 219-266 · Zbl 1038.94502 [5] Casazza P. G., Christensen O., “Perturbation of Operators and Application to Frame Theory”, J. Fourier Anal. Appl., 3 (1997), 543-557 · Zbl 0895.47007 [6] Casazza P. G., Kutyniok G., “Frames of Subspaces”, Contemp. Math., 345, 1998, 87-114 · Zbl 1058.42019 [7] Casazza P. G., Kutyniok G., Li S., “Fusion Frames and distributed processing”, Appl. comput. Harmon. Anal., 57:2 (2004), 219-266 · Zbl 1258.42029 [8] Christensen O., An Introduction to Frames and Riesz Bases, Birkhäuser, 2016 · Zbl 1348.42033 [9] Diestel J., Sequences and series in Banach spaces, Springer-Verlag, New York, 1984 · Zbl 0542.46007 [10] Duffin R. J., Schaeffer A. C., “A class of nonharmonic Fourier series”, Trans. Amer. Math. Soc., 72:1 (1952), 341-366 · Zbl 0049.32401 [11] Faroughi M. H., Ahmadi R., “Some Properties of C-Frames of Subspaces”, J. Nonlinear Sci. Appl., 1:3 (2008), 155-168 · Zbl 1171.42017 [12] Feichtinger H. G., Werther T., “Atomic Systems for Subspaces”, Proceedings SampTA (Orlando, FL, 2001), 163-165 [13] Găvruţa P., “On the duality of fusion frames”, J. Math. Anal. Appl., 333 (2007), 871-879 · Zbl 1127.46016 [14] Hansen F., Pečarić J., Perić I., “Jensens Operator inequality and its converses”, Math. Scand., 100 (2007), 61-73 · Zbl 1151.47025 [15] Heuser H., Functional Analysis, John Wiley, New York, 1991 · Zbl 0721.26001 [16] Kadison R., Singer I., “Extensions of pure states”, American Journal of Math., 81 (1959), 383-400 · Zbl 0086.09704 [17] Khayyami M., Nazari A., “Construction of Continuous g-Frames and Continuous Fusion Frames”, Sahand Comm. Math. Anal., 4:1 (2016), 43-55 · Zbl 1413.42057 [18] Li D., Leng J., “On Some New Inequalities for Fusion Frames in Hilbert Spaces”, Math. Ineq. Appl., 20:3 (2017), 889-900 · Zbl 1377.42036 [19] Matković M., Pečarić J., Perić I., “A variant of Jensen”s Inequality of Mercer’s Type For Operators with Applications”, Linear Algebra Appl., 418 (2006), 551-564 · Zbl 1105.47017 [20] Najati A., Faroughi M. H., Rahimi A., “g-frames and stability of g-frames in Hilbert spaces”, Methods of Functional Analysis and Topology, 14:3 (2008), 305-324 · Zbl 1174.46012 [21] Najati A., Rahimi A., “Generalized frames in Hilbert spaces”, Bull. Iranian Math. Soc., 35:1 (2009), 97-109 · Zbl 1180.41027 [22] Sadri V., Rahimlou G., Ahmadi R., Zarghami Farfar R., “Construction of g-fusion frames in Hilbert spaces”, Inf. Dim. Anal. Quan. Prob. (IDA-QP), 2019 (to appear) · Zbl 1454.42029 [23] Sun W., “G-Frames and G-Riesz bases”, J. Math. Anal. Appl., 326 (2006), 437-452 · Zbl 1129.42017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.