Symmetric numerical ranges of four-by-four matrices

Abstract

Numerical ranges of matrices with rotational symmetry are studied. Some cases in which symmetry of the numerical range implies symmetry of the spectrum are described. A parametrized class of $4\times 4$ matrices $K\left(a\right)$ such that the numerical range $W\left(K\left(a\right)\right)$ has fourfold symmetry about the origin but the generalized numerical range $W_{K{\left(a\right)}^{\ast }}\left(K\left(a\right)\right)$ does not have this symmetry is included. In 2011, Tsai and Wu showed that the numerical ranges of weighted shift matrices, which have rotational symmetry about the origin, are also symmetric about certain axes. We show that any $4\times 4$ matrix whose numerical range has fourfold symmetry about the origin also has the corresponding axis symmetry. The support function used to prove these results is also used to show that the numerical range of a composition operator on Hardy space with automorphic symbol and minimal polynomial $z^4-1$ is not a disk.