The supremum of linear operators for the ∗-order

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H. In 1978, Drazin introduced a partial order, on B(H). For A, if and only if A∗A=A∗B and AA∗=BA∗. Given A, B∈B(H), let be the least upper bound (supremum) for A and B with respect to the ∗-order. In this paper, we present some necessary and sufficient conditions for which exists, and give an explicit representation of (if exists).