Characterizations of Herglotz-Nevanlinna functions using positive semi-definite functions and the Nevanlinna kernel in several variables

In this paper, we give several characterizations of Herglotz-Nevanlinna functions in terms of a specific type of positive semi-definite functions called Poisson-type functions. This allows us to propose a multidimensional analogue of the classical Nevanlinna kernel and a definition of generalized Nevanlinna functions in several variables. Furthermore, a characterization of the symmetric extension of a Herglotz-Nevanlinna function is also given. The subclass of Loewner functions is also discussed, as well as an interpretation of the main result in terms of holomorphic functions on the unit polydisk with non-negative real part.