The authors study Nevanlinna functions of class \({\mathcal N}^\infty_k\) which have \(k\) poles of nonpositive type in the upper half-plane and allow an irreducible representation \(N(z)= c(z)^m N_0(z)+ p(z)\), where \(c(z):= (z- z^*_0)\).
The paper has five sections. The first gives an introduction to symmetric operators in Pontryagin spaces and Nevanlinna functions. Self-adjoint operators with cyclic elements in a Pontryagin space are studied in the second section. The main results of the paper are to be found in section 3, where the authors clarify the connections between some types of Nevanlinna functions, symmetric operators, self-adjoint operators and the corresponding Pontryagin/Hilbert spaces. Section 4 is devoted to the analytic and operator characterizations of the functions \(N(z)\) and \(\widehat N(z)\). Section 5 contains applications of this theory to Bessel operators, which motivates the entire study.