In theory, if a linear operator $P$ in a Hilbert space $H$ is self-adjoint, we can decompose it as $Pu=\sum_i \lambda_i <\phi_i,u>\phi_i$, where $\phi_i$ is the eigenfunction of $P$. And we can further obtain that $P^ku=\sum_i \lambda_i^k <\phi_i,u>\phi_i$. It is important for us because it means that we can ignore the spectral components with small eigenvalues for $P^k$ with large $k$.
But I don't know if there is a similar decomposition for non-self-adjoint operators. I knew that we can always use the SVD to get $Pu=\sum_i \lambda_i <\psi_i,u>\phi_i$, but we cannot get the result like $P^ku=\sum_i \lambda_i^k <\psi_i,u>\phi_i$. I only knew some results on the the self-adjoint/normal operators or finite-dimensional matrices. Is there any result related to general non-self-adjoint operators?