Uniqueness of commuting compact approximations

Holmes, Richard B.; Scranton, Bruce E.; Ward, Joseph D.

doi:10.1090/S0002-9947-1975-0380480-0

Uniqueness of commuting compact approximations
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by Richard B. Holmes, Bruce E. Scranton and Joseph D. Ward PDF

Trans. Amer. Math. Soc. 208 (1975), 330-340 Request permission

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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