In this paper, the notion of rank−k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for > 0, the notion of Birkhoff-James approximate orthogonality sets for −higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural general...

The classical numerical range of a quadratic operator is an elliptical disk. This result is extended to different kinds of generalized numerical ranges. In particular, it is shown that for a given quadratic operator, the rank-k numerical range, the essential numerical range, and the q-numerical range are elliptical disks; the c-numerical range is a sum of elliptical disks, and the Davis-Wieland...

There has been considerable interest in studying inclusion regions for numerical ranges. It is in fact very useful in knowing inclusion regions for W (A). For example, it is well known (see [4, Chapter 1]) that W (A) ⊆ IR if and only if A = A∗; W (A) ⊆ [0,∞) if and only if A is positive semidefinite; andW (A) ⊆ (0,∞) if and only if A is positive definite. Moreover, Ando [1] (see also [2]) showe...

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.

Linear dependence of two Hilbert space operators is expressed in terms of equality in modulus of certain sesquilinear and quadratic forms associated with the operators. The forms are based on generalized numerical ranges.