what is a spectral function?

Question

My knowledge in spectral theory is very limited, but lately I heard talking about the spectral function of an operator and how it's important. By curiosity I tried to look for a definition and a meaning of it on the internet but unfortunally I found nothing only spectral functions of some specific operators but no general definition ! even in some books about spectral theory I search for this mathematical object but in vain.

Can any one give me a reference where I can find such definition or maybe the definition itself ? My second question is why such function is important and what does it mean? Thank you for your time.

Answer 1

This is linked with the spectral theorem. There are multiple formulations of this powerful theorem, one is the following :

Spectral theorem (multiplication operator form) :

Let $\mathcal{H}$ be a separable Hilbert space and $A\in\mathcal{L}\left(\mathcal{H}\right)$, that is, $A$ is a linear bounded mapping $\mathcal{H}\to\mathcal{H}$. Then there is a measure space $\left(\Omega,\mu\right)$ with $\mu\left(\Omega\right)<+\infty$, there is an unitary mapping $U:\mathcal{H}\to L^2\left(\Omega,\mu\right)$ and there is a bounded function $F:\Omega\to\mathbb{C}$ such that $$UAU^{-1}f(x)=F(x)f(x)\quad\quad\forall f\in L^2\left(\Omega,\mu\right),\forall x\in\Omega.$$

$\mu$ is called the spectral measure and $F$ the spectral function. In a certain way, we can diagonalize $A$ on another space as we do for linear operators in finite dimension : $A$ is said to be a multiplication in $L^2$. $L^p$ spaces are well knowed and studied, especially for the case $p=2$ (it is an Hilbert space in this case), so a multiplication on it is easier to deal with rather than an operator on an abstract Hilbert space $\mathcal{H}$.

A very good reference to begin is the book "Methods of modern mathematical physics" by Reed & Simon. The above theorem is in the first volume.