Operator Convex Functions of Several Variables
Frank HansenUniversity of Copenhagen, Denmark
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  for functions of one variable.
Cite this article
Frank Hansen, Operator Convex Functions of Several Variables. Publ. Res. Inst. Math. Sci. 33 (1997), no. 3, pp. 443–463DOI 10.2977/PRIMS/1195145324