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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})^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$. 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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