Document Type : Research Paper
1 Department of Mathematics, Faculty of Science, University of Hormozgan, P.O.Box 7916193145, Bandar Abbas, Iran.
2 Department of Mathematics, Faculty of Science, University of Maragheh, P.O.Box 55136-553, Maragheh, Iran.
Let $K$ be a bounded operator. $K$-frames are ordinary frames for the range $K$. These frames are a generalization of ordinary frames and are certainly different from these frames. This research introduces a new concept of bases for the range $K$. Here we define the $K$-orthonormal basis and the $K$-Riesz basis, and then we describe their properties. As might be expected, the $K$-bases differ from the ordinary ones mentioned in this article.
 A. Askari Hemmat, A. Safapour and Z. Yazdani Fard, Coherent Frames, Sahand Commun. Math. Anal., 11 (2018), pp. 1-11.
 T. Bemrose, P.G. Casazza, K. Grochenig, M.C. Lammers and R.G. Lynch, Weaving frames, OAM, 10 (2016), pp. 1093-1116.
 P.G. Casazza and G. Kutyniok, Frames of subspaces, in: Wavelets, Frames and Operator Theory (College Park, MD, 2003), Contemp. Math. 345, Amer. Math. Soc., Providence, RI, (2004), pp. 87-113.
 O. Christensen, Frames and bases, Birkhauser, 2008.
 I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. phys., 27 (1986), pp. 1271-1283.
 S.G. Deepshikha and L.K. Vashisht, Weaving $K$-frames in Hilbert spaces, Results. Math, 73 (2018), pp. 81-100.
 R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), pp. 413-415.
 R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.
 P.A. Fillmore and J.P. Williams, On Operator Ranges, Adv. in Math., 7 (1971), pp. 254-281.
 D. Gabor, Theory of communication. Part 1: The analysis of information, J. IEE, London, 93 (1946), pp. 429-457.
 L. Gavruta, Frames for operators, App. and Comp. Harm. Anal., 32 (2012), pp. 139-144.
 M. Jia and Y.C. Zhu, Some results about the operator perturbation of a $K$-frame, Results. Math, 73 (2018), pp. 138-148.
 M. Nouri, A. Rahimi and SH. Najafzadeh, Some results on controlled $K$-frames in Hilbert spaces, Int. J. of Anal. and App., 16 (2018), pp. 62-74.
 G. Ramu and P.S. Johnson, Frame operators of $K$-frames, SeMA Journal, 73 (2016), pp. 171-181.
 M. Rashidi-Kouchi, Frames in super Hilbert modules, Sahand Commun. Math. Anal., 9 (2018), pp. 129-142.
 W. Sun, g-frames and g-Riesz bases, J. Math. Anal. Appl., 322 (2006), pp. 437-452.
 X. Xiao, Y. Zhu and L. Gavruta, Some properties of $K$-frames in Hilbert spaces, Results. Math, 63 (2013), pp. 1243-1255.