We consider linear operators $T$ on a Hilbert space which satisfy the polynomial operator equation $p(T)=A$, where $p$ is a polynomial with complex coefficients and $A$ is a normal operator which is assumed to be either reductive or unitary. We calculate spectral characteristics of $T$: the multiplicity of the spectrum, the “disc”, the lattice of invariant subspaces, and others. The main example of the $T$'s considered is given by the weighted substitution operator $Tf=\varphi (f\circ\omega)$ on $L^2(X,\nu)$, where $\omega$ is a periodic automorphism of a measure space $(X,\nu)$ and $\varphi\in L^\infty(X,\nu)$.