The star order on C∗-modular operators

Abstract

By Moore-Penrose properties and block matrix forms of C∗-modular operators we prove that
T ⩽∗ S is equivalent to T ⩽∗ S that define ordering relation, when T and S have closed ranges, we give an
explicit formula for Moore-Penrose product of S† and T, in the case it is idempotent.