Positive approximants

Abstract

Let T = B + i C T = B + iC with B = B ∗ , C = C ∗ B = {B^\ast },C = {C^\ast } and let δ ( T ) \delta (T) denote the the distance of T T to the set of nonnegative operators. We find upper and lower bounds for δ ( T ) \delta (T) . We prove that if P P is any best approximation for T T among nonnegative operators then P ≤ B + ( ( δ ( T ) ) 2 − C 2 ) 1 / 2 P \leq B + ((\delta (T))^2 - C^2)^{1/2} . Provided B ≥ 0 B \geq 0 or T T is normal we characterize those T T which have a unique best approximation among the nonnegative operators. If T T is normal we characterize its best approximating nonnegative operators which commute with it. We characterize those T T for which the zero operator is the best approximating nonnegative operator.