The reverse order rule (AB)† = B†A† for the Moore-Penrose inverse is established in several equivalent forms. Results related to other generalized inverses are also proved.

In this paper, using some block-operator matrix techniques, we give the necessary and sufficient conditions for the reverse order law for {1, 2, 3} and {1, 2, 4}−inverses of bounded operators on Hilbert spaces. Furthermore, we present new equivalent conditions for the reverse order law for the Moore-Penrose inverse. AMS classification: 15A09

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 $RPQ$†</...