I would like to verify my intuition behind the spectral theorem in infinite dimensions. For the moment I am putting bounded/unbounded and domain issues aside.
In finite dimensions the spectral theorem says that for any self-adjoint/Hermitian matrix $A$, $$A = U^{-1}\Lambda U $$ where $\Lambda$ is a diagonal matrix consisting of the eigenvalues of $A$ and $U$ is a change of basis matrix. The interpretation here is that $U$ maps to a basis that is a direct sum of eigenspaces. In each eigenspace the the action of $A$ is multiplication by the appropriate eigenvalue. Thus we may write $$A = \sum_i^n \lambda_i P_i \tag{1}$$ where $P_i$ is the projection onto the eigenspace corresponding to $\lambda_i$.
Moving to infinite dimensions, let $A$ be a self-adjoint operator on some Hilbert space $H$. Unlike the finite dimensional case, in general the spectrum of $A$:
- Is no longer countable.
- Contains elements that are not eigenvalues (i.e. the continuous part of the spectrum).
To address the first point, the sum in (1) is now made into an integral: $$A = \int_{\sigma(A)} \lambda dP_\lambda. \tag{2}$$ For points in the spectrum of $A$ that are eigenvalues, the integral (2) is to be interpreted similarly to (1). That is, we project onto the eigenspace spanned by $\lambda$ and the action of $A$ on this subspace is to multiply by $\lambda$.
Making sense of (2) for points in the continuous spectrum addresses the second point above. The operator $P_\lambda$ for $\lambda$ in the continuous spectrum is the zero projector. To account for the contribution of the continuous spectrum we make sense of (2) in a limiting sense similar to how the Lebesgue integral is constructed. That is, we consider Borel sets $\Omega$ containing an approximate eigenvalue and interpret $P_\Omega$ as projecting onto an approximate eigenspace where the action of $A$ is approximately multiplication by $\lambda$. Its contribution to (2) is obtained as $\Omega \rightarrow \{\lambda\}$ in some sense, again similar to how we use simple functions to construct the Lebesgue integral and then take a limit.
I know the above is a bit handwav and can be made more precise using distribution functions $P_\lambda = P((-\infty, \lambda])$ to define a Stieltjes integral, but my goal is to build an intuition for now. Is my intuition of the spectral theorem and the integral (2) correct?