Differential Representations of CAP Operators Associated with Quasi-Definite Moment Functionals

In this paper, we discuss several recent results on the differential representation of CAP operators in more general context with the notion of quasi-definite moment functionals. Some results obtained in [1] are improved. The explicit forms of adjoint operators of differential operators are formulated and applied to characterize symmetric differential operators, and also a necessary condition for quasi-definite moment functionals to be finite type is investigated in terms of solution of eigenvalue problem, which gives a motivation to characterize finite type quasi-definite moment functionals in terms of eigenvalue problems.