Pedersen--Takesaki operator equation in Hilbert $C^*$-modules

Abstract

We extend a work of Pedersen and Takesaki by giving some equivalent conditions for the existence of a positive solution of the so-called Pedersen-Takesaki operator equation XHX = K in the setting of Hilbert C * -modules. It is known that the Douglas lemma does not hold in the setting of Hilbert C * -modules in its general form. In fact, if E is a Hilbert C * -module and A, B ∈ L(E ), then the operator inequality BB * ≤ λAA * with λ > 0 does not ensure that the operator equation AX = B has a solution, in general. We show that under a mild orthogonally complemented condition on the range of operators, AX = B has a solution if and only if BB * ≤ λAA * and R(A) ⊇ R(BB * ). Furthermore, we prove that if L(E ) is a W * -algebra, A, B ∈ L(E ), and R(A * ) = E , then BB * ≤ λAA * for some λ > 0 if and only if R(B) ⊆ R(A). Several examples are given to support the new findings.