## Involve: A Journal of Mathematics

• Involve
• Volume 11, Number 5 (2018), 803-826.

### 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 $4×4$ matrices $K(a)$ such that the numerical range $W(K(a))$ has fourfold symmetry about the origin but the generalized numerical range $WK(a)∗(K(a))$ 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 $z4−1$ is not a disk.

#### Article information

Source
Involve, Volume 11, Number 5 (2018), 803-826.

Dates
Revised: 30 July 2017
Accepted: 3 September 2017
First available in Project Euclid: 12 April 2018

https://projecteuclid.org/euclid.involve/1523498545

Digital Object Identifier
doi:10.2140/involve.2018.11.803

Mathematical Reviews number (MathSciNet)
MR3784028

Zentralblatt MATH identifier
06866585

#### Citation

Burnett, Shelby L.; Chandler, Ashley; Patton, Linda J. Symmetric numerical ranges of four-by-four matrices. Involve 11 (2018), no. 5, 803--826. doi:10.2140/involve.2018.11.803. https://projecteuclid.org/euclid.involve/1523498545

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