# Operator Convex Functions of Several Variables

### Frank Hansen

University of Copenhagen, Denmark

## Abstract

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.

## Cite this article

Frank Hansen, Operator Convex Functions of Several Variables. Publ. Res. Inst. Math. Sci. 33 (1997), no. 3, pp. 443–463

DOI 10.2977/PRIMS/1195145324