Indefinite matrix inequalities via matrix means (Q2046122)

Let \(\mathbb{M}_{n}\) be the algebra of \(n \times n \) complex matrices and \(J \in \mathbb{M}_{n}\) denote a selfadjoint involution (that is, \(J^{\ast} = J = J ^{-1}\)), which endows \(\mathbb{C}^{n}\) with an (indefinite) inner product \([\cdot,\cdot]\) given by \([x,y] = \langle J x,y\rangle\) (\( x, y \in \mathbb{C}^{n}\)); \(\langle \cdot,\cdot\rangle \) denotes the usual inner product on \(\mathbb{C}^{n}\). The \(J\)-adjoint \( A ^{ \natural} \) of a matrix \(A\in \mathbb{M}_{n}\) is defined by \([A ^{ \natural}x, y] = [x, A y]\) (\(x, y \in\mathbb{C}^{n}\)) and \( A \) is said to be \(J\)-selfadjoint if \( A ^{ \natural}= A\). Given \(J\)-selfadjoint matrices \(A , B\), the \(J\)-order \(A \leq^{J} B\) is defined by \([Ax, x]\leq [Bx, x]\) (\(x \in\mathbb{C}^{n}\)). Let a real-valued function \(f\) be operator monotone on \((0 , \infty)\) (that is, for every \(n=1,2,\dots,\) and \( A,B \in \mathbb{M}_{n} \), inequality \(f(A ) \leq f(B)\) holds whenever \(A \leq B \) with \(f(A)\) and \(f(B)\) defined by the Riesz-Dunford integral). For a positive operator monotone function \(f\) on \((0, \infty)\), having \(f(1) = 1\), and \(J\)-selfadjoint matrices \(A,B\) with spectra in \((0 , \infty)\) and \(sp(A^{-1}B) \subset (0 , \infty)\), the authors of the paper under review define the so called \(J\)-mean \(\sigma_{f}(A,B)\) of \(A\) and \( B\) by the equality \[\sigma_{f}(A,B)= A f(A^{-1}B)(= A^{1/2}f(A^{- 1/2}BA^{-1/2})A^{1/2}). \] They establish several fundamental properties of \(J\)-means such as the power monotonicity and prove an indefinite version of the Ando-Hiai inequality.