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

TL;DR

This paper characterizes when a compact operator is a scalar multiple of a positive semi-definite operator based on the spectral inclusion of products with rank-one operators, exploring conditions for broader classes of operators.

Contribution

It establishes a spectral inclusion criterion for identifying positive semi-definite multiples among compact operators and explores when this criterion extends to other operator classes.

Findings

Compact operator is a scalar multiple of positive semi-definite iff spectral inclusion holds for all rank-one operators.

Normal operators can fail to satisfy the spectral inclusion condition.

Provides conditions for other classes of operators where the spectral criterion applies.

Abstract

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