POSITIVE OPERATORS AND INTEGRAL REPRESENTATION

Abstract

In this paper, we give a new and useful criterion for the positivity of generalized functions and study positive operators on test function space of entire functions on the dual space of a nuclear space with a certain exponential growth condition. This new criterion is used to prove that every positive operator has an integral representation given by positive Borel measure, which can be characterized by integrability conditions. Moreover, this new criterion of positivity can be easily applied to operators such as the identity, the translation, the multiplication, and the convolution operators. This enable us to obtain characterization and integral representation of the associated measure. We also apply the above results to study regularity property of the solution of some quantum stochastic differential equations.