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Commutator approximants


Author: P. J. Maher
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
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Abstract: This paper deals with minimizing $ \vert\vert B - (AX - XA)\vert{\vert _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\Vert \right\Vert _p}$ denotes its norm.) The main result (Theorem 3.2) says that if $ A$ is normal and $ AB = BA$ then $ \vert\vert B - (AX - XA)\vert{\vert _p},1 \leq p < \infty $, is minimized if and for $ 1 < p < \infty $ only if, $ AX - XA = 0$; and that the map $ X \to \vert\vert B - (AX - XA)\vert\vert _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

DOI: https://doi.org/10.1090/S0002-9939-1992-1086335-6
Keywords: Commutator, von Neumann-Schatten class, Fuglede's theorem, functional calculus
Article copyright: © Copyright 1992 American Mathematical Society

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