What is operator calculus? Article by Palais.

Question

I'm reading a paper titled Morse theory on Hilbert Manifolds by R. Palais. And in the demonstration of the Morse Lemma (pg 307), he use something called operator calculus, for example he take the characteristic function $h$ of the interval $[0,\infty)$ and simply compute it on an operator $A$, giving $h(A)=P$ where $P$ is an orthogonal projection. I really didn't understand this procedure. What does it make sense to compute $h(A)$? Can someone clarify this? Or even give me a reference?

Answer 1

You should look up the spectral theorem for self adjoint operators. One version of it (there are several) states that every self adjoint operator $A$ on a separable Hilbert space is unitarily equivalent to a multiplication operator on some Hilbert space (equipped with a possibly different measure). So to be more precise, let $A$ be a self adjoint operator on $H$, then there exists a non-negative Borel measure $\mu$ and a unitary operator $U$ such that $$U: H \to L^2(\mu)$$ and $$A_{\phi}=UAU^{-1}$$

Where $(A_{\phi}f)(x)=\phi(x)f(x)$

Now you can define what is known as functional calculus. Let $g$ be a Borel function then define $g(A)$ to be:

$$g(A):=U^{-1}A_{g \circ \phi}U$$

In simple terms, if $A$ is realized as the multiplication operator by $\phi$ on $L^2(\mu)$ then $g(A)$ is simply the multiplication operator by $g \circ \phi$ on that space.

The above could be generalized to the non-separable case as well.