Uniqueness of commuting compact approximations Academic Article

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