Operator convex functions of several variables (Q1378353)

Summary: The functional calculus for functions of several variables associates to each tuple \(x= (x_1,\dots, x_k)\) of selfadjoint operators on Hilbert spaces \(H_1,\dots, H_k\) an operator \(f(x)\) in the tensor product \(B(H_1)\otimes\cdots\otimes B(H_k)\). 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 \(x\to f(x)\). As an application, we derive that functions with positive semidefinite generalized Hessian matrices of arbitrary order are operator convex. The result generalizes a theorem of \textit{F. Kraus} [Math. Z. 41, 18-42 (1936; Zbl 0013.39701)] for functions of one variable.