Positive definite functions and dual pairs of locally convex spaces Academic Article uri icon


  • Abstract. Using pairs of locally convex topological vector spaces in duality and topologies defined by directed families of sets bounded with respect to the duality, we prove general factorization theorems and general dilation theorems for operator-valued positive definite functions … Keywords: positive definite function, locally convex space, dual pair, the (strong) factorization property, dilation theory … Complex-valued positive definite functions and their associated reproducing kernel Hilbert spaces play a key role in stochastic processes and related topics. Also of key importance in this circle of ideas are the notions of positive operator from a Hilbert space into itself, and of a dilation of a positive definite function. The framework of complex-valued functions, or even functions taking values in a Hilbert space (see for instance [5]), is too restrictive for various applications, and the case of functions taking values in a locally convex …

publication date

  • January 1, 2018