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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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Commutator approximants
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by P. J. Maher PDF
Proc. Amer. Math. Soc. 115 (1992), 995-1000 Request permission

Abstract:

This paper deals with minimizing $||B - (AX - XA)|{|_p}$, where $A$ and $B$ are fixed, $B \in {\mathcal {C}_p}$, and $X$ varies such that $AX - XA \in {\mathcal {C}_p}$. (Here, ${\mathcal {C}_p}$ denotes the von Neumann-Schatten class and ${\left \| \right \|_p}$ denotes its norm.) The main result (Theorem 3.2) says that if $A$ is normal and $AB = BA$ then $||B - (AX - XA)|{|_p},1 \leq p < \infty$, is minimized if and for $1 < p < \infty$ only if, $AX - XA = 0$; and that the map $X \to ||B - (AX - XA)||_p^p,1 < p < \infty$, has a critical point at $X = V$ if and only if $AV - VA = 0$.
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 115 (1992), 995-1000
  • MSC: Primary 47B47; Secondary 47A30, 47B10, 47B15
  • DOI: https://doi.org/10.1090/S0002-9939-1992-1086335-6
  • MathSciNet review: 1086335