I am following a course in functional analysis and in the lectures we encountered the following theorem:

Theorem: Let H be a Hilbert space and $T:H\to H$ a self-adjoint and compact operator. Then: $T = \sum_{\lambda\in\sigma(T)}\lambda\pi_\lambda$

Here $\pi_\lambda$ is the orhtogonal projection on the eigenspace corresponding to $\lambda$. We also assume that $T$ is always bounded.

My question now is how can I intuitively understand this better? Does this theorem have any nice corollaries? Can someone maybe give an example for infinite dimensional Hilbert spaces and a operator satisfying the conditions?

Thanks!

## Best Answer

It really depends on your background, but I like to think about it in terms of a spectral problem. Suppose you are interested in all pairs $(\lambda, u)$ satisfying $Lu = \lambda u$ together with some boundary conditions, where $L$ is a second-order linear differential operator. Such problem arises naturally when you are studying a mechanical system and you are looking for time-harmonic solutions of the linearized problem. The spectral theorem gives you conditions guaranteeing that $\lambda$ are just eigenvalues, i.e., you don't have to worry about residual or continuous spectrum.

The classical example is the Laplacian $L = -\Delta$ with Dirichlet boundary condition acting on $H = L^2$. Keep in mind that differential operators are not bounded operators (which means they can't be compact), so you would look at the inverse of $L$ instead.