Pedersen–Takesaki operator equation and operator equation AX = B in Hilbert C⁎-modules

Abstract

Let E be a Hilbert C⁎-module and L(E) be the C⁎-algebra of adjointable operators on E. We extend a work of Pedersen and Takesaki to the setting of Hilbert C⁎-modules by giving some equivalent conditions for the existence of a positive solution X of the so-called Pedersen–Takesaki operator equation XHX=K in which H,K∈L(E). It is known that the Douglas lemma does not hold in the setting of Hilbert C⁎-modules in its general form. If 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 an 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 findings.