*-frames for operators on Hilbert modules

Document Type : Research Paper


Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Islamic Republic of Iran


$K$-frames which are generalization of frames on Hilbert spaces‎, ‎were introduced‎ ‎to study atomic systems with respect to a bounded linear operator‎. ‎In this paper‎, ‎$*$-$K$-frames on Hilbert $C^*$-modules‎, ‎as a generalization of $K$-frames‎, ‎are introduced and some of their properties are obtained‎. ‎Then some relations‎ ‎between $*$-$K$-frames and $*$-atomic systems with respect to an adjointable operator are considered and some characterizations of $*$-$K$-frames are given‎. ‎Finally perturbations of $*$-$K$-frames are discussed‎.


[1] L. Arambasic, On frames for countably generated Hilbert C*-modules, Proc. Amer. Math. Soc. 135 (2007) 469–478.
[2] A. Alijani and M.A. Dehghan, -Frames in Hilbert C*-Modules, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 73(4) (2011), 89–106.
[3] M.S. Asgari and H. Rahimi, Generalized frames for operators in Hilbert spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2) (2014), 1450013 (20 pages).
[4] H. Bolcskei, F. Hlawatsch, H. G. Feichtinger, Frame-theoretic analyssis of oversampled filter banks, IEEE Trans. Signal Process., 46 (1998), 3256–3268.
[5] B. Dastourian and M. Janfada, Frames for operators in Banach spaces via semi-inner products, Int. J. Wavelets Multiresolut. Inf. Process., 14 (3) (2016), 1650011.
[6] R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Amer.
Math. Soc., 17 (2) (1966), 413–415.
[7] J. Dun, A.C. Schae er, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341–366.
[8] N.E. Dudey Ward, J.R. Partington, A construction of rational wavelets and frames in Hardy-Sobolev space with applications to system modelling, SIAM J. Control Optim., 36(1998), 654–679.
[9] Y.C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors. J. Fourier. Anal. Appl. 9 (1) (2003) 77–96.
[10] Y.C. Eldar and T. Werther, General framework for consistent sampling in Hilbert spaces, Int. J. Wavelets Multi. Inf. Process., 3 (3) (2005), 347–359.
[11] X. Fang, J. Yu and H. Yao, Solutions to operators equation on Hilbert C*-modules, Linear Algebra Appl., 431(11)(2009), 2142–2153.
[12] H.G. Feichtinger and T. Werther, Atomic systems for subspaces, in:L. Zayed (Ed.), Proceedings SampTA 2001, Orlando, FL, 2001, 163–165.
[13] P.J.S.G. Ferreira, Mathematics for multimedia signal processing II: Discrete finite frames and signal reconstruction, In:Byrnes, J.S. (ed.) Signal processing for multimedia, IOS Press, Amsterdam, (1999), 35–54.
[14] M. Frank and D. R. Larson, A module frame concept for Hilbert C*-modules, Functional and Harmonic Analysis of Wavelets (San Antonio, TX, Jan. 1999), Contemp. Math., 247 (2000), 207–233.
[15] M. Frank and D. R. Larson, Frames in Hilbert C*-modules and C*-algebra, J. Operator theory, 48 (2002) 273–314.
[16] L. Gavrut¸a, Frames for operators, Appl. Comput. Harmon. Anal., 32 (2012) 139–144.
[17] W. Jing, Frames in Hilbert C*-modules, Ph.D. Thesis, University of Central Florida, 2006.
[18] A. Khosravi and B. Khosravi, Frames and bases in tensor products of Hilbert spaces and Hilbert C*-modules,
Proc. Indian Acod. Sci., 117 (1) (2007), 1–12.
[19] E. C. Lance, Hilbert C*-modules, University of Leeds, Cambridge University Press, London, 1995.
[20] B. Magajna, Hilbert C*-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc.,
125(3) (1997), 849–852.
[21] V. M. Manuilov, Adjointability of operators on Hilbert C*-modules, Acta Math. Univ. Comenianae, LXV (2)
(1996), 161–169.
[22] J. G. Murphy,Operator Theory and C*-Algebras, Academic Press, San Diego, 1990.
[23] M. Skeide, Generalised matrix C*-algebras and representations of Hilbert modules,Math. Proc. R. Ir. Acad.,
100(1) (2000), 11–38.
[24] M. Pawlak and U. Stadtmuller, Recovering band-limited signals under noise, IEEE Trans. Info. Theory,
42(1994), 1425–1438.
[25] T. Strohmer and R. Heath Jr., Grassmanian frames with applications to coding and communications, Appl.
Comput. Harmon. Anal., 14 (2003) 257–275.
[26] N. E. Wegge-Olsen, K-Theory and C*-Algebras-A Friendly Approach, Oxford Uni. Press, Oxford, England,
[27] X. Xiao, Y. Zhu and L. Gavrut¸a, Some properties of K-frames in Hilbert spaces, Results. Math., 63(3-4) (2013),
[28] X. Xiao, Y. Zhu, Z. Shu, M. Ding, G-frames with bounded linear operators, Rocky Mountain J. Math., 45 (2)(2015), 675–693.
[29] L. C. Zhang, The factor decomposition theorem of bounded generalized inverse modules and their topological
continuity, J. Acta Math. Sin., 23 (2007), 1413-1418.