#### Vol. 11, No. 5, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
Symmetric numerical ranges of four-by-four matrices

### Shelby L. Burnett, Ashley Chandler and Linda J. Patton

Vol. 11 (2018), No. 5, 803–826
##### 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 $4×4$ 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 $4×4$ 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.

##### Keywords
numerical range, symmetry, weighted shift matrices, composition operator
Primary: 15A60
Secondary: 47B33
##### Milestones
Received: 13 December 2016
Revised: 30 July 2017
Accepted: 3 September 2017
Published: 2 April 2018

Communicated by Chi-Kwong Li
##### Authors
 Shelby L. Burnett California Polytechnic State University San Luis Obispo, CA United States Ashley Chandler California Polytechnic State University San Luis Obispo, CA United States Linda J. Patton California Polytechnic State University San Luis Obispo, CA United States