Intertwining relations and the algebraic structure of supraposinormal operators

Let T be a bounded linear operator on a complex Hilbert space H. We say that T intertwines two bounded linear operators A and B on H if TA = BT. In particular, T is called supraposinormal if T intertwines QT⋆ and T⋆P, for some positive operators Q and P, where at least one of P or Q has a dense range. In this work, we investigate when the product of two bounded linear operators on H remains supraposinormal. We illustrate our results with concrete examples. Finally, we characterize the supraposinormility of T using the properties of absolute values and square roots of positive operators.