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

The numerical range of a quadratic operator acting on an indefinite inner product space is shown to have a hyperbolical shape. This result is extended to different kinds of indefinite numerical ranges, namely, indefinite higher rank numerical ranges and indefinite Davis-Wielandt shells.

The numerical range of a quadratic operator acting on an indefinite inner product space is shown to have a hyperbolical shape. This result is extended to different kinds of indefinite numerical ranges, namely, indefinite higher rank numerical ranges and indefinite Davis-Wielandt shells.

We verify a conjecture on the structure of higher-rank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higher-rank numerical ranges for a generic unitary matrix are given by complex polygons determined by the spectral structure of the matrix. We discuss applications of the results to quantum error correction, s...

Let Mn be the semigroup of n× n complex matrices under the usual multiplication, and let S be different subgroups or semigroups in Mn including the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Λk(A) is the rank-k numerical range and rk(A) is the rank-k numerical radius of A ∈ Mn. Multiplicative maps φ : S → Mn satisfying rk(φ(A)...

The higher rank numerical ranges of generic matrices are described in terms the components their Kippenhahn curves. Cases tridiagonal (in particular, reciprocal) 2-periodic treated more detail.

LetW (A) andWe(A) be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A1, . . . , Am) acting on an infinite dimensional Hilbert space, respectively. In this paper, it is shown that We(A) is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ {1, . . . ,m}, We(A) can be obtained as the intersection ...