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Submitted
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August 23, 2021
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Published
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2021-12-28
Abstract
We present controlled by operators generalized fusion frame in the tensor product of Hilbert spaces and discuss some of its properties. We also describe the frame operator for a pair of controlled $g$-fusion Bessel sequences in the tensor product of Hilbert spaces.
References
- P. Casazza, and G. Kutyniok, Frames of subspaces, Contemporary Math, AMS 345 (2004), pp. 87-114.
- O. Christensen, An introduction to frames and Riesz bases, Birkhauser, 2008.
- I. Daubechies, A. Grossmann, and Y. Mayer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986), no. 5, pp. 1271-1283. https://doi.org/10.1063/1.527388
- R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366. https://doi.org/10.1090/s0002-9947-1952-0047179-6
- G.B. Folland, A Course in abstract harmonic analysis, CRC Press BOCA Raton, Florida, 1995.
- P. Gavruta, On the duality of fusion frames, J. Math. Anal. Appl., 333 (2007), pp. 871-879.
- P. Ghosh and T.K. Samanta, Stability of dual $g$-fusion frame in Hilbert spaces, Methods Funct. Anal. Topol., 26 (2020), no. 3, pp. 227-240. https://doi.org/10.31392/mfat-npu26_3.2020.04
- P. Ghosh and T.K. Samanta, Generalized atomic subspaces for operators in Hilbert spaces, Mathematica Bohemica (2021), 21 pages. https://doi.org/10.21136/mb.2021.0130-20
- P. Ghosh and T.K. Samanta, Fusion frame and its alternative dual in tensor product of Hilbert spaces, 2021 Preprint, arXiv: 2105.03094, 15 pages.
- P. Ghosh and T.K. Samanta, Generalized fusion frame in tensor product of Hilbert spaces, J. Indian Math. Soc. (accepted).
- R.V. Kadison and J.R. Ringrose, Fundamentals of the theory of operator algebras, Vol. I, Academic Press, New York, 1983.
- A. Khosravi and M.S. Asgari, Frames and bases in the tensor product of Hilbert spaces, Intern. Math. Journal, 4 (2003), no. 6, pp. 527-537.
- A. Khosravi and M. Mirzaee Azandaryani, Fusion frames and $g$-frames in tensor product and direct sum of Hilbert spaces, Appl. Anal. Discrete Math., 6 (2012), no. 2, pp. 287-303. https://doi.org/10.2298/aadm120619014k
- H. Liu, Y. Huang and F. Zhu, Controlled $g$-fusion frame in Hilbert space, Int. J. Wavelets, Multiresolution Info. Proc., 19 (2021), no. 5, article number 2150009. https://doi.org/10.1142/S0219691321500090
- G. Rahimlou, V. Sadri, and R. Ahmadi, Construction of controlled $K$-$g$-fusion frame in Hilbert spaces, U. P. B. Sci. Bull., Series A, 82 (2020), pp. 111-120.
- S. Robinson, Hilbert space and tensor products, Lecture notes, 1997.
- V. Sadri, Gh. Rahimlou, R. Ahmadi, and R. Zarghami Farfar, Generalized fusion frames in Hilbert spaces, Infin. Dimens. Anal. Quantum Probab., 23 (2020), no. 2, article number 2050015. https://doi.org/10.1142/S0219025720500150
- W. Sun, G-frames and G-Riesz bases, J. Math. Anal. and Appl., 322 (2006), no. 1, pp. 437-452.
- G. Upender Reddy, N. Gopal Reddy, and B. Krishna Reddy, Frame operator and Hilbert-Schmidt operator in tensor product of Hilbert spaces, J. Dyn. Syst. and Geom. Theor., 7 (2009), no. 1, pp. 61-70. https://doi.org/10.1080/1726037x.2009.10698563