CONCAVITY OF CERTAIN MAPS ON POSITIVE DEFINITE MATRICES AND APPLICATIONS TO HADAMARD PRODUCTS

If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ1 and Φ2 are concave maps among positive definite matrices, then the following map involving tensor products: (A,B){mapping}f[Φ1(A)-1⊗Φ2(B)]·(Φ1(A)⊗I) is proved to be concave. If Φ1 is affine, it is proved without use of positivity that the map (A,B){mapping}f[Φ1(A)⊗Φ2(B)-1]·(Φ1(A)⊗I) is convex. These yield the concavity of the map (A,B){mapping}A1-p⊗Bp (0<p≤1) (Lieb's theorem) and the convexity of the map (A,B){mapping}A1+p⊗B-p (0<p≤1), as well as the convexity of the map (A,B){mapping}(A·log[A])⊗I-A⊗log[B]. These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices. © 1979.