# 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