Vol. 11, No. 5, 2018

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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(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.

Keywords
numerical range, symmetry, weighted shift matrices, composition operator
Mathematical Subject Classification 2010
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