A SHARP ESTIMATE IN AN OPERATOR-INEQUALITY

Let H and K be Hilbert spaces, and suppose A is-an-element-of B(H) and B is-an-element-of B(K) are selfadjoint operators with dist(sigma(A), sigma(B)) is-greater-than delta > 0. It is known that for any Q is-an-element-of B(K, H) we must have pi/2 parallel-to AQ - QB parallel-to greater-than-or-equal-to delta parallel-to Q parallel-to. In this paper we give examples proving that pi/2 is sharp in this inequality.