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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A canonical trace class approximant

Authors: D. A. Legg and J. D. Ward
Journal: Proc. Amer. Math. Soc. 93 (1985), 653-656
MSC: Primary 47B10; Secondary 47A30
MathSciNet review: 776197
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Abstract: Let $ H$ be a finite-dimensional Hilbert space, $ B\left( H \right)$ the space of bounded linear operators on $ H$, and $ C$ a convex subset of $ B\left( H \right)$. If $ A$ is a fixed operator in $ B\left( H \right)$, then $ A$ has a unique best approximant from $ C$ in the $ {C_P}$ norm for $ 1 < p < \infty $. However, in the $ {C_1}$ (trace) norm, $ A$ may have many best approximants from $ C$. In this paper, it is shown that the best $ {C_p}$ approximants to $ A$ converge to a select trace class approximant $ {A_1}$ as $ p \to 1$. Furthermore, $ {A_1}$ is the unique trace class approximant minimizing $ \sum\nolimits_{i = 1}^n {{S_i}\left( {A - B} \right)\operatorname{ln }{S_i}\left( {A - B} \right)} $ over all trace class approximants $ B$. The numbers $ {S_i}\left( T \right)$ are the eigenvalues of the positive part $ \left\vert T \right\vert$ of $ T$.

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