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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Positive approximants

Author: Richard Bouldin
Journal: Trans. Amer. Math. Soc. 177 (1973), 391-403
MSC: Primary 47A65; Secondary 41A65
MathSciNet review: 0317082
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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.

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Article copyright: © Copyright 1973 American Mathematical Society