Löwner inequality of indefinite type Original Research Article

A selfadjoint involutive matrix J provides Cn with an (indefinite) inner product [x,y]≡left angle bracketJx,yright-pointing angle bracket. For a pair of J-selfadjoint matrices A,B, the J-order relation image is defined as [Ax,x]greater-or-equal, slanted[Bx,x] for all x. We will show that if A,B are J-selfadjoint matrices such that all eigenvalues of A,B are real and contained in an interval (α,β) then, for any operator monotone function f(t) on (α,β), the matrices f(A),f(B) are well defined by the Riesz–Dunford integral andimageWhen J=I and image on (0,∞), this is the classical Löwner inequality.