Document Type: Research Paper


Department of Mathematics, Faculty of Mathematics and Statistics, University of Birjand, Birjand, Iran.


In this paper, the special attention is given to the  product of two  modular operators, and when at least one of them is EP, some interesting results is made, so the equivalent conditions are presented  that imply  the product of operators is EP. Also, some conditions are provided, for which the reverse order law is hold. Furthermore, it is proved  that $P(RPQ)$ is idempotent, if $RPQ$ has closed range, for orthogonal projections $P,Q$ and $R$.


Main Subjects

[1] S. Campbell and C.D. Meyer, Continuity properties of the Drazin pseudo inverse, Linear Algebra and Its Applications, 10 (1975), pp. 77-83.

[2] S. Campbell and C.D. Meyer, EP operators and generalized inverses, Canadian Math. Bull., 18 (1975), pp. 327-333.

[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 and 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 (2007), pp. 1242-1246.

[5] D.S. Djordjevic and N.C. Din‡cic, Reverse order law for the Moore-Penrose inverse,‡ J. Math. Anal. Appl., 361 (2010), pp. 252-261.

[6] J. Farokhi-ostad and M. Mohammadzadeh Karizaki, The reverse order law for EP modular operators, J. Math. Computer Sci. 16 (2016), pp. 412-418.

[7] M. Frank, Geometrical aspects of Hilbert C*-modules, Positivity, 1999, pp. 215-243.

[8] M. Frank, Self-duality and C*-reflexivity of Hilbert C*-modules, Z. Anal. Anwendungen, 1990, pp. 165-176.

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

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

[11] M. Mohammadzadeh Karizaki and D.S. Djordjevic, Commuting C*— modular operators, Aequationes Mathematicae 6 (2016), pp. 1103-1114.

[12] M. Mohammadzadeh Karizaki, M. Hassani, and M. Amyari, Moore-Penrose inverse of product operators in Hilbert C*-modules, Filomat, 8 (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 (2012), pp. 189-196.

[15] G.J. Murphy, C*-algebras and operator theory, Academic Press Inc., Boston, MA, 1990.

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

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

[18] 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 (2016), pp. 53- 60.

[19] Q. Xu and L. Sheng, Positive semi-definite matrices of adjointable operators on Hilbert C*-modules, Linear Algebra Appl. 428 (2008), pp. 992-1000.