Continuous families of vectors that are orbits of a unitary group

We characterize functions u from the real line into a Hilbert space that are the orbits of a unitary group {U(t)}t∈R; that is, u(t)=U(t)u(0), for all real t. One of the characterizations is that u be the Fourier transform of a certain type of vector-valued measure Z; we then use our characterizations to construct Z from u.