Numerical Ranges of the product of Operators

TL;DR

This paper characterizes the numerical range of the product of two operators with line segment numerical ranges, showing it as a convex hull of elliptical disks related to the spectrum, and extends known results to self-adjoint operators.

Contribution

It provides a new geometric description of the numerical range of operator products and establishes conditions for set equality, extending previous results to self-adjoint operators.

Findings

Containment region equals convex hull of elliptical disks

Conditions for set equality in numerical ranges

Extension of results to self-adjoint operators

Abstract

We study containment regions of the numerical range of the product of operators $A$ and $B$ such that $W(A)$ and $W(B)$ are line segments. It is shown that the containment region is equal to the convex hull of elliptical disks determined by the spectrum of $AB$, and conditions on $A$ and $B$ for the set equality holding are obtained. The results cover the case when $A$ and $B$ are self-adjoint operators extending the previous results on the numerical range of the product of two orthogonal projections.