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$g$-frames and $g$-frame sequences in Hilbert spaces. (English) Zbl 1198.42032
Let $U$ and $V$ be two complex Hilbert spaces, let $\{V_j\}_{j\in J}$ be a sequence of closed subspaces of $V,$ where $J\subset\Bbb Z,$ and let $L(U,V_j)$ be the collection of all bounded linear spaces from $U$ to $V_j.$ For a $g$-frame $\{\Lambda_j:\Lambda_j\in L(U,V_j)\}_{j\in J}$ for $V$ with respect to $\{V_j\}_{j\in J}$ (i.e. there exist $A,B>0$ such that $$A\Vert f\Vert^2\leq \sum_{j\in J}\Vert \Lambda_jf\Vert^2\leq B\Vert f\Vert^2$$ for all $f\in V)$ the authors find some relations between operators $$S: f\to \sum_{j\in J}\Lambda^*_j\Lambda_jf,\quad Q: \{g_j\}_{j\in J}\to \sum_{j\in J}\Lambda_j^*g_j$$ and $A,B;$ moreover necessary and sufficient conditions for a $g$-frame in terms of $Q$ are given. Further the authors define a $g$-frame sequence $\{\Lambda_j\}_{j\in J}$ for $U$ as a $g$-frame for $$W=\overline{\{\sum_{j\in J_1}\Lambda_j^*g_j \,\text{for any finite} \,J_1\subset J \,\text{and any} \,g_j\in V_j,\: j\in J\}}.$$ They discuss that definition and consider the stability of a $g$-frame sequence under perturbation.

42C15General harmonic expansions, frames
46C99Inner product spaces, Hilbert spaces
Full Text: DOI
[1] Duffin, R. J., Schaeffer, A. C.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72, 341--366 (1952) · Zbl 0049.32401 · doi:10.1090/S0002-9947-1952-0047179-6
[2] Casazza, P. G.: The art of frame theory. Taiwanese J. Math., 4(2), 129--201 (2000) · Zbl 0966.42022
[3] Christensen, O.: Frames Riesz bases and discrete Gabor/wavelet expansions. Bull. Amer. Math. Soc., 38(3), 273--291 (2001) · Zbl 0982.42018 · doi:10.1090/S0273-0979-01-00903-X
[4] Christensen, O.: An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003 · Zbl 1017.42022
[5] Mallat, S.: A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1999 · Zbl 0998.94510
[6] Walnut, D. F.: An Introduction to Wavelet Analysis, Birkhäuser, Boston, 2002 · Zbl 0989.42014
[7] Casazza, P. G., Christensen, O.: Perturbation of operator and applications to frame theory. J. Fourier Anal. Appl., 3, 543--557 (1997) · Zbl 0895.47007 · doi:10.1007/BF02648883
[8] Christensen, O.: Operators with closed range pseudo-inverses and perturbation of frames for a subspace. Canad. Math. Bull., 42(1), 37--45 (1999) · Zbl 0938.42021 · doi:10.4153/CMB-1999-004-5
[9] Christensen, O.: Frames and pseudo-inverses. J. Math. Anal. Appl., 195, 401--414 (1995) · Zbl 0845.47002 · doi:10.1006/jmaa.1995.1363
[10] Zhu, Y. C.: q-Besselian frames in Banach spaces. Acta Mathematica Sinica, English Series, 23(9), 1707--1718 (2007) · Zbl 1123.42008 · doi:10.1007/s10114-005-0884-y
[11] Sun, W.: G-frame and g-Riesz base. J. Math. Anal. Appl., 322(1), 437--452 (2006) · Zbl 1129.42017 · doi:10.1016/j.jmaa.2005.09.039
[12] Sun, W.: Stability of g-frames. J. Math. Anal. Appl., 326(2), 858--868 (2007) · Zbl 1130.42307 · doi:10.1016/j.jmaa.2006.03.043
[13] Casazza, P. G., Kutyniok, G.: Frames of subspaces, in: Wavelets, Frames and Operator Theory, Contemp. Math., Vol. 345, Providence, RI, 2004, 87--113 · Zbl 1058.42019
[14] Asgari, M. S., Khosravi, A.: Frames and bases of subspaces in Hilbert spaces. J. Math. Anal. Appl., 308, 541--553 (2005) · Zbl 1091.46006 · doi:10.1016/j.jmaa.2004.11.036
[15] Fornasier, M.: Quasi-orthogonal decompositions of structured frames. J. Math. Anal. Appl., 289, 180--199 (2004) · Zbl 1058.46009 · doi:10.1016/j.jmaa.2003.09.041
[16] Zhu, Y. C.: Characterizations of g-frames and g-Riesz bases in Hilbert spaces. Acta Mathematica Sinica, English Series, 24(10), 1727--1736 (2008) · Zbl 1247.42031 · doi:10.1007/s10114-008-6627-0
[17] Rudin, W.: Functional Analysis, Second Edition, McGraw-Hill, New York, 1991 · Zbl 0867.46001
[18] Taylor, A. E., Lay, D. C.: Introduction to Functional Analysis, Wiley, New York, 1980 · Zbl 0501.46003
[19] Kato, T.: Perturbation Theory for Linear Operator, Springer-Verlag, New York, 1984 · Zbl 0531.47014