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 $T$.References
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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