Operator convex functions of several variables

Abstract

The functional calculus for functions of several variables associates to each tuple x = (x1, ⋯,xk) of selfadjoint operators on Hilbert spaces H1, ⋯,Hk an operator f(x) in the tensor product B(H1)⊗ ⋯ ⊗B(Hk). 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.