Derivations mapping into scattered operators

A scattered operator is a bounded linear operator with at most countable spectrum. We prove that if the range of an inner derivation on all bounded linear operators on Hilbert space is contained in the set of scattered operators, then the range is contained in the set of compact operators. As a corollary we prove that on the direct product of countably many copies of B(H), if for some quasinilpotent operator Q, the sum of Q and any quasinilpotent operator is scattered, then Q is compact.