K-Frame of multipliers in Hilbert pro-C*-module
Subject Areas : StatisticsMona Naroei Irani 1 , Akbar Nazari 2
1 - Department of Mathematics‎, ‎Kerman Branch‎, ‎Islamic‎
‎Azad University, ‎Kerman‎, ‎Iran.
2 - Department of Pure Mathematics‎, ‎Faculty of Mathematics and Computer‎, ‎Shahid Bahonar University of Kerman‎, ‎Kerman‎, ‎Iran.
Keywords: K-قاب از ضربگرها, سیستم های اتمی, واژه های کلیدی: -C*-proمدول هیلبرتی, قاب از ضربگرها,
Abstract :
abstractFor the study of atomic systems, first introduced by Feichtinger et al. Gavruta presented K-frames on Hilbert spaces. K-frames are a kind of frames in sense that the lower frame bound only holds for the elements in the range of the K, where K is a bounded linear operator in Hilbert space. C*-algebra whose topology is induced by a family of continuous C*-seminorms instead of a C*-norm is called pro-C*-algebra. Hilbert pro-C*-modules are generalizations of Hilbert spaces by allowing the inner product to take values in a pro-C*-algebra rather than in the field of complex numbers. In this paper, the sequences whose elements are adjointable operators from pro-C*-algebra into Hilbert pro-C*-module is called the sequence of multipliers. We introduce the concept atomic systems and K-frame of multipliers in Hilbert pro-C*-modules and for more information, we give an example of K-frames. We obtain a condition that sequences of multipliers is frame, also we investigate the relationship between atomic systems and K-frames with each other and the frame of multipliers. If K is a bounded operator with certain conditions then every K-frame of multipliers is a frame of multipliers in Hilbert pro-C*-module. Also, we investigate some of the properties of these concepts, such as the combination of operators with K-frames in Hilbert pro-C*-module.
[1] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Transactions of the American Mathematical Society, 72, 341-366, (1952).
[2] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, Journal of Mathematical Physics, 27, 1271-1283, (1986).
[3] W. Sun, G-frames and g-Riesz bases, Journal of Mathematical Analysis and Applications, 322, 437-452, (2006).
[4] P. G. Casazza and G. Kutyniok, Frames of subspaces, Wavelets, Contemporary Mathematics American Mathematical Society, 345, 87-113, (2004).
[5] M. Rashidi-kouchi, The study on controlled g-frames and controlled fusion frames in Hilbert C*-modules, Journal of New Researches in Mathematics, 5, 105-114, (2019).
[6] M. Frank and D. R. Larson, Frame in Hilbert C*-modules and C*-algebras, Journal of Operator Theory, 48, 273-314, (2002).
[7] A. Alijani and M.A. Dehghan, G-frames and their duals for Hilbert C*-modules, Bulletin of the Iranian Mathematical Society, 38, 3, 567-580, (2012).
[8] M. A. Dehghan and M. A. Hasankhani Fard, G-continuous frames and coorbit spaces, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, 24, 373-383, (2008).
[9] M. A. Dehghan and M. Radjabalipour, Relation between generalized frames, Matematicheskoe
Modelirovanie, 14 (5), 31-34, (2002).
[10] I. Raeburn and S.J. Thompson, countably generated Hilbert modules, the Kasparov stabilisation theorem, and frames with Hilbert modules, Proceedings of the American Mathematical Society, 131 (5), 1557-1564, (2003).
[11] M. Naroei Irani and A. Nazari, Some properties of *-frames in Hilbert modules over pro-C*-algebras, 16 (1), 105-117, (2019).
[12] M. Naroei Irani and A. Nazari, The woven frame of multipliers in Hilbert C*-modules, Communications of the Korean Mathematical Society, accepted.
[13] H. G. Feichtinger and T. Werther, Atomic systems for subspaces, Proceedings of the International Conference on Sampling Theory and Applications, Orlando, 163-165, (2001).
[14] L. Gavruta, Frames for operators, Applied and Computational Harmonic Analysis, 32, 139-144, (2012).
[15] M. Joita, Hilbert Modules Over Locally C*-Algebras, University of Bucharest Press, (2006).
[16] N. Haddadzadeh, G-frames in Hilbert pro-C*-modules, International Journal of Pure and Applied Mathematics, 105, 273-314, (2015).
[17] M. Azhini and N. Haddadzadeh, Fusion frames in Hilbert modules over pro-C*-algebras, International Journal of Industrial Mathematics, 5, 109-118, (2013).
[18] M. Joita, On frames in Hilbert modules over pro-C*-algebras, Topology and its Applications, 156 (1), 83-92, (2008).