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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A canonical trace class approximant
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by D. A. Legg and J. D. Ward PDF
Proc. Amer. Math. Soc. 93 (1985), 653-656 Request permission

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 $\left | T \right |$ of $T$.
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 93 (1985), 653-656
  • MSC: Primary 47B10; Secondary 47A30
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0776197-2
  • MathSciNet review: 776197