Partial isospectrality of a matrix pencil and circularity of the c-numerical range

Abstract

We study when functions of the eigenvalues of the pencil

Re(e−itA)=cos(t)ReA+sin(t)ImA (1)

are constant functions of t. The results are then applied to questions regarding the numerical range, the higher rank numerical range and the c-numerical range, and we derive trace type conditions for when these numerical ranges are disks centered at 0. The theory of symmetric polynomials plays an important part in the proofs.