Controlled $K$-frames in Hilbert C*-modules
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Abstract
Controlled frames have been the subject of interest because of their ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the notion of controlled $K$-frame or controlled operator frame in Hilbert $C^{*}$-modules. We establish the equivalent condition for controlled $K$-frame. We investigate some operator theoretic characterizations of controlled $K$-frames and controlled Bessel sequences. Moreover, we establish the relationship between the $K$-frames and controlled $K$-frames. We also investigate the invariance of a controlled $K$-frame under a suitable map $T$. At the end, we prove a perturbation result for controlled $K$-frame.
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References
[1] A. Najati, M. M. Saem and P. Gavruta, Frames and operators in Hilbert C∗-modules, Operators and Matrices 10 (1) (2016), 73–81. Google Scholar
[2] D. Han, W. Jing and R. Mohapatra, Perturbation of frames and Riesz bases in Hilbert C∗- modules, Linear Algebra Appl. 431 (2009), 746–759. Google Scholar
[3] D. Han, W. Jing, D. Larson and R. Mohapatra, Riesz bases and their dual modular frames in Hilbert C∗-modules, J. Math Anal. Appl. 343 (2008), 246–256. Google Scholar
[4] E. J. Candes and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities, Commun. Pure Appl. Math. 57 (2) (2004), 219–266. Google Scholar
[5] H. Bolcskei, F. Hlawatsch and H. G. Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Process. 46 (12) (1998), 3256–3268. Google Scholar
[6] I. Daubechies, A. Grossmann and Y. Meyer, Painless non orthogonal expansions, J. Math.Phys. 27 (1986), 1271–1283. Google Scholar
[7] I. Kaplansky, Algebra of type I, Ann. Math. 56 (1952), 460–472. Google Scholar
[8] L. Arambaic, On frames for countably generated Hilbert C∗-modules, Proc. Amer. Math Soc. Google Scholar
[9] (2007), 469–478. Google Scholar
[10] L. Gavruta, Frames for operators, Appl. Comput. Harmon. Anal. 32 (2012), 139–144. Google Scholar
[11] M. Frank and D. R. Larson, Frames in Hilbert C∗-modules and C∗-algebras, J. Operator Theory 48 (2002), 273–314. Google Scholar
[12] M. Nouri, A. Rahimi and Sh. Najafzadeh, Controlled K-frames in Hilbert Spaces, J. of Ramanujan Society of Math. and Math. Sc. 4 (2) (2015), 39–50. Google Scholar
[13] M. R. Kouchi and A. Rahimi, On controlled frames in Hilbert C∗-modules, Int. J. Walvelets Multi. Inf. Process. 15 (4) (2017), 1750038. Google Scholar
[14] P. Balazs, J-P. Antoine and A. Grybos, Weighted and Controlled Frames, Int. J. Walvelets Multi. Inf. Process. 8 (1) (2010), 109–132. Google Scholar
[15] P. J. S. G. Ferreira, Mathematics for multimedia signal processing II: Discrete finite frames and signal reconstruction, In: Signals Processing for Multimedia J. S. Byrnes(Ed.)(1999),35–54. Google Scholar
[16] R. J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Math.Soc. 72 (1952), 341–366. Google Scholar
[17] T. Strohmer and R. Jr. Heath, Grassmanian frames with applications to coding and communications, Appl. Comput. Harmon. Anal. 14 (2003), 257–275. Google Scholar
[18] W. Jing, Frames in Hilbert C∗-modules, Ph.D. Thesis, University of Central Frorida, (2006). Google Scholar
[19] W. Paschke, Inner product modules over B∗-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. Google Scholar
[20] X. Fang, J. Yu and H. Yao, Solutions to operator equations on Hilbert C-modules, Linear Alg. Appl. 431 (11) (2009), 2142–2153. Google Scholar
[21] Y. C. Eldar and T. Werther, General framework for consistent sampling in Hilbert spaces, Int. J. Walvelets Multi. Inf. Process. 3 (3) (2005), 347–359. Google Scholar
[22] Y. C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, J. Fourier. Anal. Appl. 9 (1) (2003), 77–96. Google Scholar