Inequalities for J-Hermitian matrices

Abstract

Indefinite versions of classical results of Schur, Ky Fan and Rayleigh-Ritz on Hermitian matrices are stated to J-Hermitian matrices, J = I r ⊕ −I n−r , 0 < r < n. Spectral inequalities for the trace of the product of J-Hermitian matrices are presented. The inequalities are obtained in the context of the theory of numerical ranges of linear operators on indefinite inner product spaces. We denote by σ ± J (A) the sets of the eigenvalues of A with eigenvectors x such that x * J x = ±1. We recall that a J -Hermitian matrix A is J -unitarily diagonalizable if and only if every eigenvalue of A belongs either to σ + J (A) or to σ − J (A). In this case, σ + J (A) (respectively, σ − J (A)) consists of r (respectively, n − r) eigenvalues. Let A be a J -Hermitian matrix whose eigenvalues α 1 · · · α r belong to σ + J (A) and α r+1 · · · α n belong to σ − J (A). The eigenvalues of A are said to not interlace if either α r > α r+1 or α n > α 1 . Otherwise, they are said to interlace.

The boundary of the Krein space tracial numerical range, an algebraic approach and a numerical algorithm

“…A numerical algorithm to draw the boundary of W J C (A) for an arbitrary A ∈ M n and C a J -Hermitian matrix is here presented. Theorem 2.1 and 1.1 of [1] provide the general principle. Our approach uses the elementary idea that ∂ W J C (A) may be traced by computing the supporting lines of W J C (A) as specified below.…”

“…So there exists a line parallel to the imaginary axis separating σ J + (A) and σ J − (A). Then, by Theorem 1.1 of [1] tr(CU HU −1 ) : U ∈ U r,n−r = (−∞, a 0 ], a 0 = n j=1 c j λ j .…”

“…This set is a connected set in the Argand plane C and satisfies the symmetry property W J C (A) = W J A (C) (see [1,7,8] for other properties). If r = n, W J C (A) is the C-numerical range of A simply denoted by W C (A).…”

The J-numerical range of a J-Hermitian matrix and related inequalities

“…which was investigated for J -Hermitian matrices A, C under certain conditions [5,18]. For J = I n , W J C (A) is called the C-numerical range of A, and is simply denoted by W C (A).…”

“…In Theorem 2.1 of [5], it has been shown that if the eigenvalues of A are not all real, and assuming that c r > c r+1 , then W J C (A) is the whole real line. Notice that having in mind the inequality c 1 > c r+s , this constraint may be removed.…”