The spectrum of the product of operators, and the product of their numerical ranges

We show that a compact operator A is a multiple of a positive semi-definite operator if and only ifσ(AB)⊆W(A)W(B)¯,for all (rank one) operators B. An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators A so that equivalence conditions hold.