Every Operator Almost Commutes with a Compact Operator

Abstract. In this note we set forth three possible definitions of the property of “almost commuting with a compact operator ” and discuss an old result of W. Arveson that says that every operator on Hilbert space has the weakest of the three properties. Finally, we discuss some recent progress on the hyperinvariant subspace problem (see the bibliogra-phy), and relate it to the concept of almost commuting with a compact operator. 1.