cK-frames and cK-Riesz bases in Hilbert spaces

Document Type : Research Paper


Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran.



In this paper, we prove some new results about cK-frames. Also, we introduce the concept of cK-Riesz basis and we provide a necessary and sufficient condition under which $F$ is a cK-Riesz basis. Finally, for the closed range operator $K \in B(\mathcal{H})$, we prove that under some conditions, $\pi_{R(K)}F$ is a cK-Riesz basis if and only if it has only one dual, where $\pi_{R(K)}$ is the orthogonal projection from $\mathcal{H}$ onto $R(K)$, i.e., the range of $K$.


[1] S.T. Ali, J.P. Antoine and J.P. Gazeau, Coherent States, Wavelets and their Generalizations, Berlin, Springer-Verlag, 2000.
[2] S.T. Ali, J.P. Antoine and J.P. Gazeau, Continuous frames in Hilbert spaces, Ann Physics., 222 (1993), 1-37. 
[3] A.A. Arefijamaal, The continuous Zak transform and generalized Gabor frames, Mediterranean Journal of Mathematics
     10 (2013), 353-365. 
[4] A.A. Arefijamaal, R.A. Kamyabi Gol, R. Raisi Tousi and N. Tavallaei, A new approach to continuous Riesz bases, J. Sci., 
     Islamic Rep. Iran, 129 (2000), 1143-1147. 
[5] A. Askari-Hemmat, M.A. Dehghan and M. Rajabalipour, Generalized frames and their redundancy, Proc. Amer. Math. 
     Soc., 129(4) (2001), 1143-1147. 
[6] J.P. Aubin,  Applied Functional Analysis, A Wiley Series of Texts, Monographs and Tracts, 2nd Edition, 2000. 
[7] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhu$\ddot{a}$ser, 2003. 
[8] O. Christensen, Frames and Bases-an Introductory Course, Applied and Numerical Harmonic Analysis, 
      Birkhu$\ddot{a}$ser, Boston, Mass, USA, 2008. 
[9] I. Daubechies, A. Grossman and Y. Meyer, Painless nonorthogonal expansions, Journal of Mathematical Physics, 27 
     (1986), 1271-1283.
[10] R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc.,
       17 (1966), 413-415. 
[11] J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341-366. 
[12] M. Fornasier and H. Rauhut, Continuous frames, function spaces, and the discretization problem, The Journal of 
        Fourier Analysis and Applications, 11(3) (2005), 245-287. 
[13] J.P. Gabardo and D. Han, Frames associated with measurable space, Adv. Comp. Math., 18(3) (2003), 127-147.  
[14] L. G$ \check{a} $vruta, Frames for operators, Appl. Comput. Harmon. Anal., 32 (2012), 139-144. 
[15] C. Heil, Wiener Amalgam Spaces in Generalized Harmonic Analysis and Wavelet Theory, Ph.D. Dissertation, University 
       of Maryland, College Park, MD, 1990. 
[16] Y. Huang and D. Hua, K-Riesz frames and the stability of K-Riesz frames for Hilbert spaces, Sci. Sin. Math., 47}(3) 
        (2017), 383-396.
[17] G. Kaiser, A Friendly Guide to Wavelets, Boston: Birkh$ \ddot{a} $user, 1994.  
[18] G. Kaiser, Quantum physics, relativity, and complex space time, North-Holland Mathematics Studies, Vol. 163, North-
        Holland Amsterdam, 1990. 
[19] S.K. Kaushik, L.K. Vashisht and S.K. Sharma, Some results concerning frames associated with measurable spaces, 
       TWMS J. Pure Appl. Math., 4(1) (2013), 52-60.
[20] M. Rahmani, On some properties of c-frames, J. Math. Research with Appl., 37(4) (2017), 466-476.
[21] A. Rahimi, A. Najati and Y.N. Dehghan, Continuous frames in Hilbert spaces, Methods of Functional Analysis and 
       Topology, 12(2) (2006), 170-182.
[22] G.H. Rahimlou, R. Ahmadi, M.A. Jafarizadeh and S. Nami, Continuous K-frames and their dual in Hilbert spaces, 
       Sahand Commun. Math. Anal., 17(3) (2020), 145-160.
[23] G.H. Rahimlou, R. Ahmadi, M.A. Jafarizadeh and S. Nami, Some properties of continuous K-frames in Hilbert spaces, 
       Sahand Commun. Math. Anal., 15(1) (2019), 169-187.
[24] A. Shekari and M.R. Abdollahpour, On some properties of K-g-Riesz bases in Hilbert spaces, Wavelets and Linear 
       Algebra, 8(2) (2022), 31-42.
[25] A.E. Taylor and D.C. Lay, Introduction to Functional Analysis, New York, John Wiley and Sons, 1980. 
[26] X. Xiao, Y. Zhu and L. G$ \check{a} $vruta, Some properties of K-frames in Hilbert spaces, Results Math., 63 (2013), 
[27] Y. Zhu, Z. Shu and X. Xiao, K-frames and K-Riesz bases in complex Hilbert spaces, Sci. Sin. Math., 48 (2018), 609-622.