Characterize operators whose reducing subspaces coincide with those of their modulus

Characterize the necessary and sufficient conditions for a bounded operator T on a separable Hilbert space H to satisfy R_T=R_{|T|}, where R_T denotes the family of reducing subspaces of T and |T|=(T^*T)^{1/2} is the modulus of T.

Background

The paper studies minimum-attaining operators on reducing subspaces and observes that, in general, an operator T and its modulus |T| need not have the same reducing subspaces. Earlier results establish one-way implications between membership of T and |T| in the class of operators that are minimum attaining on reducing subspaces, while the final section focuses on the additional structural hypothesis R_T=R_{|T|}.

Under this hypothesis, the authors prove an equivalence between T belonging to the new class and |T| belonging to its positive part, and derive a representation of T using a countable well-ordered family of positive scalars, orthogonal projections, and partial isometries. The paper subsequently proves that the equality R_T=R_{|T|} is equivalent to the polar partial isometry V in T=V|T| belonging to the von Neumann algebra W^*(|T|), thereby resolving the explicitly posed question within the paper.