In this paper, numerical ranges associated to operators on an indefinite inner product space are investigated. Boundary generating curves, corners, shapes and computer generations of these sets are studied. In particular, the MurnaghanKippenhahn theorem for the classical numerical range is generalized.Keywords: numerical range, generalized numerical range, indefinite inner product space.

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.

Some logarithmic trace inequalities involving the notions of relative entropy are reobtained from a log-majorization result. The thermodynamic inequality is generalized and a chain of equivalent statements involving this inequality and the Peierls-Bogoliubov inequality is obtained.

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