Definitizability of Certain Functions and the Existence of Eigenvectors of Unitary Operators in Pontrjagin Spaces

Let P_k^c(G) denote the set of continuous functions with k negative squares on a locally compact commutative group G. Every function f ? P_k^c(G) is definitizable in the sense that is positive definite for certain complex measures ω on G with finite support [9]. The proof of this fact was base on a result of M. A. Naimark about common nonpositive eigenvectors of commuting unitary operators in a Pontrjagin space. It is the aim of this note to prove without any use of the theory of Pontrjagin spaces the definitizability of functions f ? P_k^c(G) which are of polynomial growth. In Section 3 we show, how the definitizability of functions f ? P_k^c(G) can be used to prove the existence of common non-positive eigenvectors of commuting unitary operators in a Pontrjagin space.