Let be a Hilbert ⁎-module and be the ⁎-algebra of adjointable operators on . We extend a work of Pedersen and Takesaki to the setting of Hilbert ⁎-modules by giving some equivalent conditions for the existence of a positive solution X of the so-called Pedersen–Takesaki operator equation in which . It is known that the Douglas lemma does not hold in the setting of Hilbert ⁎-modules in its general form. If , then the operator inequality ⁎⁎ with does not ensure that the operator equation has a solution, in general. We show that under an orthogonally complemented condition on the range of operators, has a solution if and only if ⁎⁎ and ⁎. Furthermore, we prove that if is a ⁎-algebra, , and ⁎, then ⁎⁎ for some if and only if . Several examples are given to support the findings.

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