A Note on Sequential Product of Quantum Effects

The quantum effects for a physical system can be described by the set E(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. For A, B is an element of E(H), let A circle B = A(1/2)BA(1/2) be the sequential product and let A * B = (AB + BA)/2 be the Jordan product of A, B is an element of E(H). The main purpose of this note is to study some of the algebraic properties of effects. Many of our results show that algebraic conditions on A circle B and A * B imply that A and B have 3 x 3 diagonal operator matrix forms with I<(R(A))over bar>>boolean AND<(R(B))over bar> as an orthogonal projection on closed subspace <(R(A))over bar>>boolean AND<(R(B))over bar> being the common part of A and B. Moreover, some generalizations of results known in the literature and a number of new results for bounded operators are derived.