Löwner inequality of indefinite type (Q1827499)

The author generalizes a well-known theorem of Löwner which states that for any two selfadjoint matrices \(A\), \(B\), \(A\geq B\geq 0\Rightarrow A^{1/2}\geq B^{1/2}\). A complex selfadjoint involution \(J\) defines an inner product as follows: \([x,y]:=\langle Jx,y\rangle\). Then a matrix \(A\) is called \(J\)-selfadjoint if \(JA\) is selfadjoint, and for two \(J\)-selfadjoint matrices \(A\), \(B\) the \(J\)-order relation is defined as follows: \(A\geq_JB\) if \([Ax,x]\geq [Bx,x]\) for all \(x\), or \(JA\geq JB\). Furthermore, a continuous real-valued function \(f\) on an interval \(I\) is said to be operator-monotone on \(I\) if the following property holds for all \(n\): for any two selfadjoint \(n\times n\) matrices \(A\), \(B\) all of whose eigenvalues are real and lie in \(I\), \(A\geq B\Rightarrow f(A)\geq f(B)\). Then Löwner's theorem states that the square-root function is operator-monotone on \((0,\infty)\). The main result of the paper is that for any two \(J\)-selfadjoint matrices \(A\), \(B\) all of whose eigenvalues are real and lie in \(I\), \(A\geq_J B\Rightarrow f(A)\geq_J f(B)\) for all operator-monotone functions \(f\) on \(I\). The author also discusses some corollaries involving the \(J\)-modulus of a square matrix.