Numerical ranges of Foguel operators revisited

TL;DR

This paper investigates the numerical ranges of Foguel operators, specifically those with scalar multiples of the identity, revealing that their numerical ranges are not elliptical disks as previously conjectured.

Contribution

The paper explicitly describes the numerical range of Foguel operators with scalar multiples of the identity, disproving the earlier conjecture that these ranges are elliptical disks.

Findings

Numerical range of $F_{aI}$ is explicitly described.

Contradicts previous conjecture that $W(F_{aI})$ is elliptical.

Provides new insights into the geometry of Foguel operators' numerical ranges.

Abstract

The Foguel operator is defined as $F_T=\begin{bmatrix}S^* & T \\ 0 & S\end{bmatrix}$, where $S$ is the right shift on a Hilbert space $\mathcal H$ and $T$ can be an arbitrary bounded linear operator acting on $\mathcal H$. Obviously, the numerical range $W(F_0)$ of $F_T$ with $T=0$ is the open unit disk, and it was suggested by Gau, Wang and Wu in their LAA'2021 paper that $W(F_{aI})$ for non-zero $a\in\mathbb C$ might be an elliptical disk. In this paper, we described $W(F_{aI})$ explicitly and, as it happens, it is not.