Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Science, University of Ibn Tofail, B.P. 133, Kenitra, Morocco.

Abstract

In this work, we introduce a new concept of integral $K$-operator frame for the set of all adjointable operators from a Hilbert $C^{\ast}$-module $\mathcal{H}$ to  itself denoted by $End_{\mathcal{A}}^{\ast}(\mathcal{H})$.  We give some properties relating to some constructions of integral $K$-operator frames and to operators preserving  integral $K$-operator frame and we establish some new results.

Keywords

###### ##### References
[1] S.T. Ali, J.-P Antoine and J.-P. Gazeau, Continuous frames in Hilbert space, Ann. Phy., {222} (1993), pp. 1-37.

[2] A. Alijani and M.A. Dehghan, $ast$-Frames in Hilbert $mathcal{C}^{ast}$-modules, U.P.B. Sci. Bull., Series A, {73} (2011), pp. 89-106.

[3] L. Arambasic, On frames for countably generated Hilbert $mathcal{C}^{ast}$-modules, Proc. Amer. Math. Soc., {135} (2007), pp. 469-478.

[4] O. Christensen, An Introduction to Frames and Riesz Bases, Brikh$ddot{a}$user, Boston, 2003.

[5] J.B. Conway, A Course In Operator Theory, Amer. Math. Soc., Rhode Island, 2000.

[6] K.R. Davidson, $mathcal{C}^{ast}$-algebra by example, Amer. Math. Soc., Rhode Island, 1996.

[7] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.

[8] M. Frank and D.R. Larson, Frames in Hilbert $mathcal{C}^{ast}$-modules and $mathcal{C}^{ast}$-algebras, J. Operator Theory, 48 (2002), pp. 273-314.

[9] J.P. Gabardo and D. Han, Frames associated with measurable spaces, Adv. Compu. Math., 18 (2003), pp. 127-147.

[10] L. G$breve{a}$vruta, Frames for operators, Appl. Comput. Harmon. Anal., 32 (2012), pp. 139-144.

[11] I. Kaplansky, Modules over operator algebras, Amer. J. Math., 75 (1953), pp. 839-858.

[12] A. Khosravi and B. Khosravi, Frames and bases in tensor products of Hilbert spaces and Hilbert $mathcal{C}^{ast}$-modules, Proc. Indian Acad. Sci. Math. Sci., 117 (2007), pp. 1-12.

[13] A. Najati, M. Mohammadi Saem and P. G$breve{a}$vruta, Frames and operators in Hilbert $mathcal{C}^{ast}$-modules, Oper. Matrices, 10 (2016), pp. 73-81.

[14] W.L. Paschke, Inner product modules over $B^{ast}$-algebras, Trans. Amer. Math. Soc., 182 (1973), pp. 443-468.

[15] M. Rossafi and A. Akhlidj, Perturbation and stability of operator Frame for $End_{mathcal{A}}^{ast}(mathcal{H})$,
Math-Recherche and Applications, 16(1) (2018), pp. 65-81.

[16] M. Rossafi, F. Chouchene and S. Kabbaj, Integral Frame in Hilbert $C^{ast}$-module, arXiv:2005.09995v2 [math.FA] 30 Nov 2020.

[17] M. Rossafi and S. Kabbaj, $K$-operator Frame for $End_{mathcal{A}}^{ast}(mathcal{H})$, Asia Mathematika, 2(2) (2018), pp. 52-60.

[18] M. Rossafi and S. Kabbaj, Operator Frame for $End_{mathcal{A}}^{ast}(mathcal{H})$, J. Linear. Topological. Algebra., 8(2) (2019), pp. 85-95.

[19] K. Yosida, Functional Analysis, Springer-Verlag, Germany, 1980.

[20] L.C. Zhang, The factor decomposition theorem of bounded generalized inverse modules and their topological continuity, J. Acta Math. Sin., 23 (2007), pp. 1413-1418.