Sufficient Condition for AB = BA

Let H, B(H) be separable complex Hilbert space and the set of all bounded linear operators on H, respectively. The elements of E(H) = {A ∈ B(H) : 0 ≤ A ≤ I} are called quantum effects. For A,B ∈ E(H), the sequential product of A and B is A ◦B = A 12BA 12 and the Jordan product of A and B is A ∗ B = AB+BA2. Many of results show that alge-braic conditions on A ◦B or A ∗B imply that AB = BA [?]-[?]. There are some questions as following. 1. Can we get that Ai, 1 ≤ i ≤ n are commutative if sequential product An ◦ An−1