Matrix convexity of functions of two variables
It is shown that the separately matrix convex functions in two variables on a rectangle \(I \times J\) containing (0,0) are the functions satisfying the following inequality for all projections \(P\) and \(Q\) and all Hermitian matrices \(A\) and \(B\) with spectrum contained in \(I\) and \(J\) resp.: \[ (P \otimes Q) f(PAP,QBQ) (P \otimes Q) \leq (P \otimes Q) f(A,B) (P \otimes Q). \] It is also shown that the two notions matrix convexity on \(I \times I\) and diagonal matrix convexity on \(I \times I\) coincide.
scientific article; zbMATH DE number 540062