Operator Convex Functions of Several Variables
Frank Hansen
University of Copenhagen, Denmark
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Abstract
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.
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