Two-moment characterization of spectral measures on the real line

Abstract In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked following general question. When is a positive operator measure projection valued? A version this question formulated in terms moments was posed Pietrzycki Stochel (2021, Functional Analysis 280, 109001). Let T be self-adjoint operator, let F Borel semispectral on real line with compact support. For which integers $p< q$ do equalities $T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)$ , $k=p, imply that spectral measure? present paper, we completely solve second problem. The answer affirmative if $p$ odd $q$ even, negative otherwise. case $(p,q)=(1,2)$ closely related to intrinsic noise solved by several including Kruszyński de Muynck, as well Ylinen. counterpart problem concerning multiplicativity unital linear maps $C^*$ -algebras also provided.