Isospectrality and matrices with concentric circular higher rank numerical ranges

Abstract

We characterize under what conditions n×n Hermitian matrices A1 and A2 have the property that the spectrum of cos⁡tA1+sin⁡tA2 is independent of t (thus, the trigonometric pencil cos⁡tA1+sin⁡tA2 is isospectral). One of the characterizations requires the first ⌈n2⌉ higher rank numerical ranges of the matrix A1+iA2 to be circular disks with center 0. Finding the unitary similarity between cos⁡tA1+sin⁡tA2 and, say, A1 involves finding a solution to Lax's equation.