For a physics application, I would like to be able to compute the eigenvalues of the linear operator (acting on the Hilbert space $\ell^2$) given by an infinite matrix of the form

$\begin{bmatrix} a_0 & a_1 & a_2 & a_3 & \dots\cr a_1 & a_0 & a_1 & a_2 & \dots\cr a_2 & a_1 & a_0 & a_1 & \dots\cr a_3 & a_2 & a_1 & a_0 & \dots\cr \vdots & \vdots & \vdots &\vdots & \ddots \end{bmatrix}$

(where the $a_i \in \mathbb{R}$). Some Googling tells me these (or at least, the finite dimensional analogue) are known as symmetric Toeplitz matrices, but I'm having trouble finding answers to the following questions:

What conditions must there be on the $a_i$ for the above operator to be compact (so we can apply the spectral theorem). Is it enough for $\sum_{i=0}^{\infty} a_i^2$ to be finite? (This question might be easy, it's just been a long time since I've done any functional analysis).

In the case that the above operator is compact (and so the spectral theorem applies), is there any good way to approximate its eigenvalues/eigenvectors? In particular, do the eigenvalues and eigenvectors of the upper-leftmost $N$ by $N$ finite submatrix (which is also symmetric Toeplitz) "approach" the eigenvalues/eigenvectors of the operator in any sense.

Is there a known closed form for the eigenvalues/eigenvectors? I really doubt this, but since these operators "kind of" look like circulant matrices (whose eigenvectors do have a fairly nice closed form), perhaps there is some really subtle roots-of-unity trick?

EDIT: I can't seem to get the LaTeX for the above matrix to display properly on my computer. If other people are having the same problem, it is supposed to look like this:

```
a_0 a_1 a_2 a_3 ...
a_1 a_0 a_1 a_2 ...
a_2 a_1 a_0 a_1 ...
a_3 a_2 a_1 a_0 ...
. . . . .
. . . . .
```