Eigenvalues of Unitary Matrices

I am considering the standard equation for a unitary transformation

$$\alpha^* = U \alpha U^{-1}$$,

where $$\alpha$$ is an arbitrary linear operator and $$U$$ is a unitary matrix. Since in quantum mechanics observables correspond to linear operators, I am wondering if there is some way of measuring an observable and then extrapolating back to surmise that the corresponding linear operator was transformed by some particular unitary in the course of its dynamics, and then to determine the eigenvalues of that unitary matrix.

Let me know if it is unclear what I mean and I can explain further.

• "In the course of its dynamics" is a paraphrase of "time-evolved by the exponential of a hamiltonian"? – Cosmas Zachos Jun 9 at 20:27
• @Cosmas Zachos Yes you could put it that way. Let me know if anything else is unclear. – Tom Jun 10 at 16:52