Excess of continuous $K$-$g$-frames and some other properties

Document Type : Research Paper

Authors

1 Department of mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Islamic Republic of Iran.

2 Department of mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.

3 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Islamic Republic of Iran.

10.22072/wala.2019.109025.1230

Abstract

In this paper, we study the excess of continuous $K$-$g$-frames and give some results about this notion.
Also, we extend the concept of atomic system to continuous version and study its relations by continuous $K$-$g$-frames. Indeed, we give some equivalent  characterizations for continuous $K$-$g$-frames.  As well as, the relationship of a continuous $K$-$g$-frame and the range of operator $K$ will be verified. Finally, we study the induced $cK$-frames by continuous $K$-$g$-frames.

Keywords


[1] M.R. Abdollahpour and M.H. Faroughi, Continuous $G$-Frames in Hilbert spaces, Southeast Asian Bull. Math., 32 (2008), 1-19.
[2] E. Alizadeh, A. Rahimi, E. Osgooei and M. Rahmani, Continuous $K$-$G$-frames in Hilbert spaces, Bull. Iran. Math. Soc., 45(4) (2019), 1091-1104.
[3] M. Bownik, Continuous Frames and the Kadison-Singer Problem. In: Antoine JP., Bagarello F., Gazeau JP. (eds) Coherent States and Their Applications, Springer Proc. Phys., 205 (2018), 63-88.
[4] O. Christensen, Frames and Bases: An Introductory Course, Birkhauser, Boston, 2008.
[5] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal Expansions, J. Math. Phys., 27(5) (1986), 1271-1283.
[6] R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Am. Math. Soc., 17(2) (1966), 413-415.
[7] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Am. Math. Soc., 72 (1952), 341-366.
[8] L. Gavruta, Atomic decomposition for operators in reproducing kernel Hilbert spaces, Math. Rep., Buchar., 17(3) (2015), 303-314.
[9] L. Gavruta, Frames for operators, Appl. Comput. Harmon. Anal., 32(1) (2012), 139-144.
[10] D. Hua and Y. Hung, $K$-$g$-frames and stability  of $K$-$g$-frames in Hilbert spaces, J. Korean  Math. Soc., 53(6) (2016), 1331-1345.
[11] A. Najati, M.H. Faroughi and A. Rahimi, $G$-frames and stability of $g$-frames in Hilbert spaces, Methods Funct. Anal. Topol., 14 (2008),  271-286.
[12] M. Rahmani, Characterization of continuous $g$-frames via operators, arXiv:1804.04615v2 [math.FA].