Let $\mathcal{H}$ be a reproducing kernel Hilbert space of analytic functions over the unit disk $\mathbb{D}$. Let $T_z:~f\mapsto zf$ be the coordinate multiplier on $\mathcal{H}$. Suppose that $A$ is a zero set for $\mathcal{H}$ and $I_A$ is the invariant subspace for $T_z$ determined by $A$. Under some mild conditions, we prove that the spectrum of the compression of $T_z$ on $\mathcal{H}\ominus I_A$ is the closure of $A$. Since the reproducing kernel Hilbert spaces considered in this paper cover many classical spaces of analytic functions, such as Hardy space, Bergman space, some weighted Bergman spaces and etc., our approach is a uniform one, mainly based on operator theory, which covers the specific result in specific space.