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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.

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