# are spectral measures characterized by their moments?

On a Hilbert space $$\cal H$$, consider an essentially self-adjoint operator $$A\colon Dom(A)\to {\cal H}$$, and a vector $$\psi\in\bigcap_{n=1}^\infty Dom(A^n)$$. Without further assumptions, can we say that the associated spectral measure $$\mu_\psi$$ is the unique Borel measure satisfying $$\langle \psi,A^n\psi\rangle = \int\lambda^n\mu_{\psi}(d\lambda),$$ for all $$n\ge 0$$ ?

The answer is no. There are random variables with all moments finite, whose distributions are not determined by their moments (e.g. the log-normal distribution).

Let $$X$$ be such a random variable and let $$\mu$$ be its distribution. Then the operator of multiplication by $$x$$ on the space $$L^{2}(\mathbb{R},\mu)$$ has $$\psi:= 1$$ (the constant function) in the domain of all of its powers, but the spectral measure in this case is $$\mu$$ and it is not determined by moments.