Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Mathematics Science and Statistics, University of Birjand, Birjand 9717851367, Iran.

2 Department of Basic sciences, Birjand University of Technology, Birjand 9719866981, Iran.

10.22130/scma.2020.130093.821

Abstract

In this paper, number of properties, involving invertibility, existence of Moore-Penrose inverse and etc for modular operators with the same ranges on Hilbert $C^*$-modules  are presented. Natural decompositions of operators with closed range enable us to find some relations of the product of operators with  Moore-Penrose inverses under the condition that they have  the same ranges  in Hilbert $C^*$-modules. The triple reverse order law and the mixed reverse order law in the special cases are also given. Moreover, the range property and Moore-Penrose inverse are illustrated.

Keywords

[1] R. Bouldin, Closed range and relative regularity for products, J. Math. Anal. Appl., 61, (1977), pp. 397-403.

[2] R. Bouldin, The product of operators with closed range, Tohoku Math. J., 25(2), (1973), pp. 359-363.

[3] C.Y. Deng and H.K. Du, Representations of the Moore-Penrose inverse for a class of 2 by 2 block operator valued partial matrices, Linear Multilinear Algebra, 58, (2010), pp. 15-26.
 
[4] D.S. Djordjevic, Further results on the reverse order law for generalized inverses, SIAM J. Matrix Anal. Appl., 29(4), (2007), pp. 1242-1246.

[5] J. Farokhi-Ostad and A.R. Janfada, Products of EP operators on Hilbert $C^*$-Modules, Sahand Commun. Math. Anal., 10, (2018), pp. 61-71.
 
[6] J. Farokhi-Ostad and A.R. Janfada, On closed range $C^*$-modular operators, Aust. J. Math. Anal. Appl., 15(2), (2018), pp. 1-9.

[7] S. Izumino, The product of operators with closed range and an extension of the reverse order law, Tohoku Math. J., 34(2), (1982), pp. 43-52.

[8] M. Jalaeian, M. Mohammadzadeh Karizaki and M. Hassani, Conditions that the product of operators is an EP operator in Hilbert $C^*$-module, Linear Multilinear Algebra, 68(10), (2020), pp. 1990-2004.

[9] E.C. Lance, Hilbert $C^*$-Modules, LMS Lecture Note Series 210, Cambridge Univ. Press, 1995.

[10] M. Mohammadzadeh Karizaki, Antisymmetric relations of operators which satisfy specified conditions, Funct. Anal. Approx. Comput., 10(3), (2018), pp. 15-20

[11] M. Mohammadzadeh Karizaki and D.S. Djordjevic, Commuting $C^*$- modular operators, Aequationes mathematicae, 90(6), pp. 1103-1114.

[12] M. Mohammadzadeh Karizaki, M. Hassani and M. Amyari, Moore-Penrose inverse of product operators in Hilbert $C^*$-modules, Filomat, 30(13), (2016), pp. 3397-3402.

[13] M. Mohammadzadeh Karizaki, M. Hassani, M. Amyari and M. Khosravi, Operator matrix of Moore-Penrose inverse operators on Hilbert $C^*$-modules, Colloq. Math., 140, (2015), pp. 171-182.
 
[14] M.S. Moslehian, K. Sharifi, M. Forough and M. Chakoshi, Moore-Penrose inverse of Gram operator on Hilbert $C^*$-modules, Studia Math., 210(2), (2012), 189-196.

[15] G.J. Murphy, $C^*$-algebras and operator theory, Academic Press Inc., Boston, MA, 1990.
 
[16] Z. Niazi Moghani, M. Khanehgir and M. Mohammadzadeh Karizaki, Explicit solution to the operator equation $AXD + FX^*B = C$ over Hilbert $C^*$-modules, J. Math. Anal. Appl., 10(1), (2019), pp. 52-64.

[17] K. Sharifi, EP modular operators and their products, J. Math. Anal. Appl., 419, (2014), pp. 870-877.

[18] K. Sharifi, The product of operators with closed range in Hilbert $C^*$-modules, Linear Algebra Appl., 435, (2011), pp. 1122-1130.

[19] K. Sharifi and B. Ahmadi Bonakdar, The reverse order law for Moore-Penrose inverses of operators on Hilbert $C^*$-modules, Bull. Iran. Math. Soc., 42(1), (2016), pp. 53-60.

[20] M. Vosough and M.S. Moslehian, Operator and Matrix Equations, Ph.d thesis, Ferdowsi University of Mashhad, (2017).

[21] M. Vosough and M.S. Moslehian, Solutions of the system of operator equations $BXA=B=AXB=AXB$ via $*$-order, Electron. J. Linear Algebra, 32, (2017), pp. 172-183.
 
[22] Q. Xu and L. Sheng, Positive semi-definite matrices of adjointable operators on Hilbert $C^*$-modules, Linear Algebra Appl., 428, (2008), pp. 992-1000.