I am trying to find some literature on infinite matrices because I want to know how to get the eigenvalues of infinite matrices. Seriously, it seems there are very few references available. Can someone tell me how to determine the spectrum of infinite matrices?



Eigenvalues of infinite matrices
One simple example with a special matrix, which has somehow "a continuum" as eigenvalue...  

Consider some function $ f(x) = K + ax + bx^2 + cx^3 + ... $ having a nonzero radius of convergence. Then think of the infinite matrix of the form
$$ \small \begin{bmatrix}
 K & . & . & . & \cdots \\\ 
 a & K & . & . & \cdots \\\ 
 b & a & K & . & \cdots \\\ 
 c & b & a & K & \cdots \\\ 
 \vdots & \vdots & \vdots& \vdots & \ddots 
\end{bmatrix} $$ 
From the properties of finite matrices we would expect, that *K* is an eigenvalue. But consider a type of an infinite vector


$$ V(x) = [1,x,x^2,x^3,x^4,\ldots ] $$ with a scalar parameter $x$ from the range of convergence, then 
$$ V(x) \cdot F = f(x) \cdot V(x) $$
This means also: any vector $V(x)$ is an eigenvector of the matrix ***F*** and corresponds to the eigenvalue $f(x)$. If now $f(x)$ is entire, for instance the exponential function $ f(x)=\exp(x)$, then any value from the complex plane (except $0$ because $\exp(x)$ is never $0$) "is an eigenvalue" of ***F*** contradicting the "naive" extrapolation from the finite truncation of the matrix ...


