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 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. 
Chi-Kwong Li, Ming-Cheng Tsai, Kuo-Zhong Wang, and Ngai-Ching Wong
math.FA