Uniqueness of commuting compact approximations
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Let be an infinite dimensional complex Hilbert space, and let (resp. ) be the algebra of all bounded (resp. compact) linear operators on . It is well known that every has a best approximation from the subspace . 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 . In particular, many familiar operators are shown to have zero as a unique commuting best approximant.