Positive approximants

Abstract:

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