2021/11/24 by Eskandari, R., Fang, X., Moslehian, M. S. +1
#46L05 #46L08 #47A62 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2111.12601
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 \mathscrE is a Hilbert C^*-module and A, B ∈ L(\mathscr E), then the operator inequality B B^*≤ 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 \mathscr R(A) ⊇ \mathscr R(BB^*). Furthermore, we prove that if L(\mathscr E) is a W^*-algebra, A,B∈ L(\mathscr E), and \mathscr R(A^*)=\mathscr E, then BB^*≤λAA^* for some λ>0 if and only if \mathscr R (B)⊆ \mathscr R(A). Several examples are given to support the new findings.