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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Uniqueness of commuting compact approximations

Authors: Richard B. Holmes, Bruce E. Scranton and Joseph D. Ward
Journal: Trans. Amer. Math. Soc. 208 (1975), 330-340
MSC: Primary 47A65
MathSciNet review: 0380480
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Abstract: Let $ H$ be an infinite dimensional complex Hilbert space, and let $ \mathcal{B}(H)$ (resp. $ \mathcal{C}(H)$) be the algebra of all bounded (resp. compact) linear operators on $ H$. It is well known that every $ T \in \mathcal{B}(H)$ has a best approximation from the subspace $ \mathcal{C}(H)$. The purpose of this paper is to study the uniqueness problem concerning the best approximation of a bounded linear operator by compact operators. Our criterion for selecting a unique representative from the set of best approximants is that the representative should commute with $ T$. In particular, many familiar operators are shown to have zero as a unique commuting best approximant.

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Article copyright: © Copyright 1975 American Mathematical Society