Operator Convex Functions of Several Variables
1997 年 33 巻 3 号 p. 443-463
詳細
1997 年 33 巻 3 号 p. 443-463
The functional calculus for functions of several variables associates to each tuple x=(x1, …, xk) of selfadjoint operators on Hilbert spaces H1, …, Hk an operator f(x) in the tensor product B(H1)⊗…⊗B(Hk). 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.
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