Document Type: Research Paper

Authors

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

Abstract

‎In this paper‎, ‎by using the sequence of multipliers‎, ‎we introduce frames with algebraic bounds in Hilbert pro-$ C^* $-modules‎. ‎We investigate the relations between frames and $ \ast $-frames‎. ‎Some properties of $ \ast $-frames in Hilbert pro-$ C^* $-modules are studied‎. ‎Also‎, ‎we show that there exist two differences between $ \ast $-frames in Hilbert pro-$ C^* $-modules and Hilbert $ C^* $-modules‎. ‎Finally‎, ‎dual $ \ast $-frames in Hilbert pro-$ C^* $-modules are presented‎.

Keywords

Main Subjects

[1] A. Alijani, M. A. Dehghan, ∗-frames in Hilbert C^∗-modules, U.P.B. Sci. Bull. (Ser A). 73 (4) (2011), 89-106.

[2] L. Alizadeh, M. Hassani, On frames for countably generated Hilbert modules over locally C^∗-algebras, Commun. Korean Math. Soc. 33 (2) (2018), 527-533.

[3] P. G. Casazza, G. Kutyniok, Frames of subspaces, in wavelets, frames, and operator theory, Contemp. Math. 345 (2004), 87-113.

[4] I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271- 1283.

[5] R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366.

[6] M. Frank, D. R. Larson, A module frame concept for Hilbert C^∗-modules, Functional and Harmonic Analysis of Wavelets, Contemp. Math. 247 (2000), 207-233.

[7] M. Frank, D. R. Larson, Frames in Hilbert C^∗-modules and C∗-algebras, J. Operator Theory. 48 (2002), 273-314.

[8] N. Haddadzadeh, G-frames in Hilbert pro-C^∗-modules, Inter. J. Pure. Appl. Math. 105 (4) (2015), 727-743.

[9] D. Han, W. Jing, R. M. Mohapatra, Structured parseval frames in Hilbert C^∗-modules, Contemp. Math. 414 (2006), 275-287.

[10] W. Jing, Frames in Hilbert C∗-modules, Ph.D. Thesis, University of Central Florida, Orlando, USA, 2006.

[11] M. Joita, Hilbert Modules Over Locally C∗-Algebras, University of Bucharest press, 2006.

[12] M. Joita, On frames in Hilbert modules over pro-C∗-algebras, Topol. Appl. 156 (2008), 83-92.

[13] W. L. Paschke, Inner product modules over B∗-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468.

[14] I. Raeburn, S. J. Thompson, Countably generated Hilbert modules, the Kasparov stabilisation theorem and frames with Hilbert modules, Proc. Amer. Math. Soc. 131 (5) (2003), 1557-1564.

[15] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006), 437-452.

[16] Yu. I. Zhuraev, F. Sharipov, Hilbert modules over locally C^∗-algebras, arXiv: math, 2001.