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

Abstract

Recently, indefinite versions of classical inequalities of Schur, Ky Fan and Rayleigh-Ritz on Hermitian matrices have been obtained for J -Hermitian matrices that are J -unitarily diagonalizable, J = I r ⊕ (−I s ), r, s > 0. The inequalities were obtained in the context of the theory of numerical ranges of operators on indefinite inner product spaces. In this paper, the subject is revisited, relaxing the constraint of the matrices being J -unitarily diagonalizable.

On the Courant–Fischer theory for Krein spaces

“…Necessary and sufficient conditions for a J-Hermitian matrix A to satisfy the above semiboundedness were provided in [11]. To state them, we consider the generalized eigenspace X λ = {x ∈ C n :…”

“…In [1], Ando presented a Löwner inequality of indefinite type, and in [2,11] indefinite versions of well known matrix inequalities were given. These inequalities were obtained in the context of the theory of numerical ranges of operators in spaces with an indefinite inner product, a subject which is being investigated by some authors (see, e.g.…”

Krein spaces numerical ranges and their computer generation

“…In this case, W J (A) is a line or the union of two half-lines [13]. Next, we assume that A ∈ M n is nonessentially J−Hermitian.…”