Document Type : Research Paper


1 LaSMA Laboratory Department of Mathematics Faculty of Sciences, Dhar El Mahraz University Sidi Mohamed Ben Abdellah, Fez, Morocco.

2 Laboratory of Partial Differential Equations, Spectral Algebra and Geometry, Department of Mathematics, Faculty Of Sciences, University of Ibn Tofail, Kenitra, Morocco.

3 Department of Mathematics, University of Central Florida, Orlando, FL., 32816, USA.


In the present paper, we introduce  the generalized inverse operators, which have an exciting role in operator theory. We establish Douglas' factorization theorem type  for  the Hilbert pro-$C^{\ast}$-module.We introduce the notion of atomic system and $K$-frame in the Hilbert pro-$C^{\ast}$-module and study their relationship. We also demonstrate some properties of the $K$-frame by using Douglas' factorization theorem.Finally  we demonstrate that the sum of two $K$-frames in a Hilbert pro-$C^{\ast}$-module with certain conditions is once again a $K$-frame.


Main Subjects

1. M. Azhini and N. Haddadzadeh, Fusion frames in Hilbert modules over pro-$C^{*}$-algebras, Int. J. Industrial Math, (2013), pp. 109-118.
2. R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Am. Math. Soc., 17.2 (1966), pp. 413-415.
3. R.J. Duffin and A.C. Schaeffer, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.
4. X. Fang, M.S. Moslehian and Q. Xu, On majorization and range inclusion of operators on Hilbert $C^{*}$-modules, Linear and Multilinear Algebra, 2018.
5. X. Fang, J. Yu and H. Yao, Solutions to operator equations on Hilbert $C^{*}$-modules, Linear Algebra Appl., (2009), pp. 2142–2153.
6. M. Fragoulopoulou, An introduction to the representation theory of topological $\ast$-algebras, Schriftenreihe, Univ. Münster, 48 (1988), pp. 1-81.
7. M. Fragoulopoulou, Tensor products of enveloping locally $C^{*}$- algebras, Schriftenreihe, Univ. Münster, (1997), pp. 1-81.
8. D. Gabor, Theory of communications, Journal of the Institution of Electrical Engineers, 93 (26), (1946), pp. 429–457.
9. L. Găvruţa, Frames for operators, Appl. Comput. Harmon. Anal., 32.1 (2012), pp. 139-144.
10. A. Inoue, Locally $C^{*}$-algebra, Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics, 25.2 (1972), pp. 197-235.
11. M. Joita, Hilbert modules over locally C*-algebras, Editura Universităţii din Bucureşti, 2006.
12. M. Joita, On frames in Hilbert modules over pro-$C^{*}$-algebras, Topol. Appl., 156 (2008), pp. 83-92.
13. E.C. Lance, Hilbert $C^{*}$-modules, London Math. Soc, 210. Univ. Press, Cambridge, 1995.
14. A. Mallios, Topological algebras: Selected Topics, North Holland, Elsevier, 2011.
15. N.C. Phillips, Inverse limits of C*-algebras, J. Operator Theory, 19 (1988), pp. 159-195.
16. N.C. Phillips, Representable K-theory for $\sigma$ -$C^{\ast}$-algebras, K-Theory, 3 (1989), pp. 441-478.
17. Yu. I. Zhuraev and F. Sharipov, Hilbert modules over locally $C^{*}$-algebras, arXiv preprint math/0011053, 2000.