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

Abstract

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

11 pages