The multiplication version of spectrum theorem says that for a bounded self-adjoint operator $A$ on $H$, there exists measures $\{\mu_n\}$ and a unitary operator $U : H\rightarrow \oplus_{n=1}^{n} L^2(\mathbb{R},\mu_n)$ s.t. $$UAU^{-1}(\phi_i)=\lambda\phi_i$$
where $H$ can be decomposed into a set of cyclic vector $\oplus A^jv_i$.
It can be easily deduced that when $A$ is $n\times n$ matrix, then let $\mu_n=\delta(x-\lambda_i)$, then we can get spectrum decomposition on $\mathbb{R}^n$ and H can be decomposed into $\oplus v_i$ where $v_i$ is the eigenvector. However, for general $H$,it is quite puzzling for me what $A^jv_i$ represents, it looks like it mimics the form of Jordan block, and is there a link between these cyclic vector and approximate eigenvector
