  We show that a compact operator $A$ is a multiple of a positive semi-definite
operator if and only if $$ \sigma(AB) \subseteq \overline{W(A)W(B)},
\quad\text{for all (rank one) operators $B$}. $$ 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.
