Generalized inverses and operator equations

In this paper we obtain conditions under which the operator equations of the types AX = C and $\text{AXA}{}^1\text{= C}$ have hermitian and nonnegative definite solutions; here A is assumed to be relatively invertible. In addition we obtain some properties of generalized inverses of operators. Lastly we pose some conjectures; one of them is that the set of all nonzero relatively invertible operators is not connected.