The module factorization of operators on Hilbert space (Q1263055)

Let \({\mathcal E}\) be a nest of projections on a complex Hilbert space \({\mathcal H}\), and let \(E\to \tilde E\) be a left continuous order homomorphism of \({\mathcal E}\). Let \({\mathcal U}=\{X\in B({\mathcal H}):\quad XE=\tilde EXE\quad for\quad all\quad E\in {\mathcal E}\}\) and \({\mathcal U}^{\perp}=\{X\in B({\mathcal H}):\quad \tilde EX=\tilde EXE\quad for\quad all\quad E\in {\mathcal E}\}.\) Then \({\mathcal U}\) and \({\mathcal U}^{\perp}\) are respectively Alg \({\mathcal E}\) and Alg \({\mathcal E}^{\perp}\) modules. In this article the author gives some results about factorizations of the form \(A=ST\) for an invertible operator \(A\in B({\mathcal H})\), where \(S\in {\mathcal U}\), \(S^{*^{-1}}\in {\mathcal U}^{\perp}\), \(T\in {\mathcal U}^{\perp}\), \(T^{*^{-1}}\in {\mathcal U}\), and \({\mathcal U}\) is a module. When A is positive, it is shown that \(A=S^*S\) with \(S\in {\mathcal U}\) and \(S^{*^{-1}}\in {\mathcal U}^{\perp}\) if and only if there is a unitary operator U such that \(U\tilde E_ N=E_{A^{1/2}N}U\) for every \(E_ N\in {\mathcal E}\).