AFA A Glance Ahead to Spectral Theory

Section 4.7 A Glance Ahead to Spectral Theory

In Chapter 8, we will discuss the spectral theory for linear operators in detail. Here we present some of the main results on eigenvalues and spectral bounds. As we discussed the two standard examples of the second derivative operator and the integral operator as examples of unbounded versus compact operators, respectively, a discussion of their spectra fits very naturally in this chapter. The proofs of the spectral results are given later in Chapter 8.

The integral operator \(K\) is our example of a compact operator. It satisfies the Hilbert-Schmidt theorem:

Theorem 4.7.1.

Let \(A:H\rightarrow H\) be a linear, symmetric, compact operator on a Hilbert space \(H\text{.}\) Then

All eigenvalues of \(A\) are real and there is at most one accumulation point at \(0\text{.}\)
The eigenvectors \(\{w_j\}\) can be chosen to form an orthonormal basis and \(A\) has a spectral representation
\begin{equation*} Au=\sum_{j=1}^\infty\lambda_j(u,w_j)w_j. \end{equation*}

The second derivative is an unbounded operator. Depending on the boundary conditions we often are in a situation where the solution operator of a Poisson equation is compact (see our previous example Subsection 4.3.1). Hence the next spectral theorem is a typical situation for a Laplace operator problem.

Theorem 4.7.2.

Let \(A:D(A)\rightarrow H, R(A)=H\) be symmetric, linear, and unbounded, and let \(A^{-1}\) exists and be compact. Then

There exists an infinite set \(\{\lambda_n\}\) of real eigenvalues with
\begin{equation*} \lim_{n\rightarrow \infty}\mid\lambda_n\mid = +\infty. \end{equation*}
The eigenvectors \(\{w_j\}\) can be chosen to form an orthonormal basis and
\begin{equation*} Au=\sum_{j=1}^\infty\lambda_j(u,w_j)w_j. \end{equation*}

Example 4.7.3.

An example for the Hilbert-Schmidt spectral theorem. Consider the simple integral operator \(K:L^2(0,L)\to L^2(0,L)\) given by

\begin{equation*} K(f) = \int_0^1 f(s) ds. \end{equation*}

Then the eigenvalue problem for \(K\) reads

\begin{equation*} \int_0^1 f(s) ds = \lambda f(x), \qquad \mbox{for all} \qquad x\in [0,1], \end{equation*}

which is only true for constant functions \(f(x)=c\text{,}\) and the eigenvalue is \(\lambda_1=1\text{.}\) The normalized eigenfunction is \(w_1 = 1\text{,}\) and these are all the eigenfunctions. Writing out the spectral representation of \(K\text{,}\) we have

\begin{equation*} K(f) = \lambda_1 (f,w_1) w_1 = \int_0^1 f(s) ds , \end{equation*}

as it should be.