  Kiukas, Lahti and Ylinen asked the following general question. When is a
positive operator measure projection valued? A version of this question
formulated in terms of operator moments was posed in a recent paper of the
present authors. Let $T$ be a selfadjoint operator and $F$ be a Borel
semispectral measure on the real line with compact support. For which positive
integers $p< q$ do the equalities $T^k =\int_{\mathbb{R}} x^k F(dx)$, $k=p, q$,
imply that $F$ is a spectral measure? In the present paper, we completely solve
the second problem. The answer is affirmative if $p$ is odd and $q$ is even,
and negative otherwise. The case $(p,q)=(1,2)$ closely related to intrinsic
noise operator was solved by several authors including Kruszy\'{n}ski and de
Muynck as well as Kiukas, Lahti and Ylinen. The counterpart of the second
problem concerning the multiplicativity of unital positive linear maps on
$C^*$-algebras is also solved.
