# Spectral representation intuitive explanation

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!

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.