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

Chi-Kwong Li, Ming-Cheng Tsai, Kuo-Zhong Wang +1

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