Operator Convex Functions of Several Variables

Abstract

The functional calculus for functions of several variables associates to each tuple $x=(x_1,\cdots ,x_k)$ of selfadjoint operators on Hilbert spaces $H_1,\cdots ,H_k$ 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 semi-definite generalized Hessian matrices of arbitrary order are operator convex. The result generalizes a theorem of Kraus [15] for functions of one variable.

A correction to this paper is available.