Operator convex functions of several variables

The functional calculus for functions of several variables associates to each type x = (x(1),...,x(k)) of selfadjoint operators on Hilbert spaces H-1...,H-k an operator f(x) in the tensor product B(H-1)x...xB(H-k). We introduce the notion of generalized Hessian matrices associated with f: Those matrices are used as the building blocks of a structure theorem for the second Freche: differential of the map x-->f(x). As an application we derive that functions with positive semi-definite generalized Hessian matrices of arbitrary order are operator convex. The result generalizes a theorem of Kraus [15] for functions of one variable.