Operator Convex Functions of Several Variables

  • Frank Hansen

    University of Copenhagen, Denmark


The functional calculus for functions of several variables associates to each tuple of selfadjoint operators on Hilbert spaces an operator in the tensor product . 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 . 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.

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