On Some Properties of K-g-Riesz Bases in Hilbert Spaces

Document Type : Research Paper

Authors

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

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

10.22072/wala.2021.535986.1341

Abstract

In this paper, we study the K-Riesz bases and the K-g-Riesz bases in Hilbert spaces. We show that for $K \in B(\mathcal{H})$, a K-Riesz basis is precisely the image of an orthonormal basis under a bounded left-invertible operator such that the range of this operator includes the range of $K$. Also, we show that $\lbrace \Lambda_i \in B(\mathcal{H}, \mathcal{H}_i ) : \, i \in I \rbrace$ is a K-g-Riesz basis for $\mathcal{H}$ with respect to $\lbrace \mathcal{H}_i \rbrace_{i \in I}$
if and only if there exists a g-orthonormal basis $\lbrace Q_i \rbrace_{i \in I}$
for $\mathcal{H}$ and a bounded right-invertible operator $U $ on $\mathcal{H}$
such that $\Lambda_i = Q_i U$ for all $i \in I$, and $R(K) \subset R(U^{*})$.

Keywords


[1] M.R. Abdollahpour and F. Bagarello,  On some properties of g-frames and g-coherent states, Nuovo Cimento, X. Ser. 
      B, 125(11), (2010), 1327-1342.
[2] M.R. Abdollahpour and A. Najati, Approximation of the inverse G-frame operator, Proc. Indian Acad. Sci., Math. Sci.,
     121(2), (2011), 143-154.
 [3] M.R. Abdollahpour and A. Najati,  Besselian G-frames and near g-Riesz bases, Appl. Anal. Discrete Math., 5, (2011), 
      259-270.
[4] M.R. Abdollahpour and A. Najati,  G-frames and Hilbert-Schmidt operators, Bull. Iran. Math. Soc., 37(4), (2011), 
     141-155. 
[5] J.P. Aubin, Applied Functional Analysis,  A Wiley Series of Texts, Monographs and Tracts, 2nd Edition, 2000. 
[6] O. Christensen, An Introduction to Frames and Riesz Bases, Birkh$ ddot{a} $user, Boston, 2003. 
[7] R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Am. Math. Soc.
     17(2), (1966),  413-415. 
[8] J. Duffin and A.C. Schaeffer,  A class of nonharmonic Fourier series, Trans. Am. Math. Soc., 72, (1952), 341-366. 
[9] L. G$ check{a} $vruta,  Frames for operators, Appl. Comput. Harmon. Anal., 32}(1), (2012), 139-144. 
[10] D. Han and D. Larson,  Frames, bases and group representations, Mem. Am. Math. Soc., 147(697), (2000), 1-110.
[11] Y. Huang and D. Hua, K-Riesz frames and the stability of K-Riesz frames for Hilbert spaces, Sci. Sin., Math., 47(3),  
       (2017), 383-396.
[12] A. Najati, M.H. Faroughi and A. Rahimi, G-frames and Stability of g-frames in Hilbert Spaces, Methods Funct. Anal. 
       Topol., 14(3), (2008), 271-286. 
[13] W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl., 322(1), (2006), 437-452. 
[14] W. Sun,  Stability of g-frames, J. Math. Anal. Appl., 326, (2007), 858-868.
[15] A.E. Taylor and D. Lay, Introduction to Functional Analysis, New York, 1980. 
[16] X. Xiao and Y. Zhu, Exact K-g-frames in Hilbert spaces, Result. Math., 72(3), (2017), 1329-1339.
[17] X. Xiao, Y. Zhu and L. G$ check{a} $vruta, Some properties of K-frames in Hilbert spaces, Result. Math., 63(3-4), 
       (2013), 1243-1255. 
[18] Y. Zhou and Y.C. Zhu,  Characterization of K-g-frames in Hilbert spaces, Acta Math. Sin., Chin. Ser., 57(5), (2014), 
       1031-1040.
[19] Y. Zhou and Y.C. Zhu,  K-g-frames and dual g-frames for closed subspaces, Acta Math. Sin., Chin. Ser., 56(5), (2013), 
        799-806. 
[20] Y. Zhu, Z. Shu and X. Xiao,  K-frames and K-Riesz bases in complex Hilbert spaces, Sci. Sin., Math., 48, (2018),  
        609-622.