Operator Convex Functions of Several Variables

The functional calculus for functions of several variables associates to each tuple 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)⊗ ⋯ ⊗B(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 Fréchet 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.