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?

closed as not a real question by Alain Valette, Qiaochu Yuan, Yemon Choi, Mark Meckes, Andy Putman Oct 16 '12 at 5:35
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One simple example with a special matrix, which has somehow "a continuum" as eigenvalue... $$ V(x) = [1,x,x^2,x^3,x^4,\ldots ] $$ 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 ... 


to present an example: if the considered matrix is the adjacency matrix of a graph, there are relatively involved graph theoretical criteria just to decide whether the spectral radius is an eigenvalue, let alone further spectral values. i think you should make yourself clearer what exactly you would like to know. 


For me, infinite matrix means an operator on $\ell^2(\mathbb N)$ (or sometimes $\ell^2(\mathbb Z)$, but usually referred to biinfinite matrix). Concerning the eigenvalues, you thus may just look at the general theory concerning operator on Hilbert spaces, as already pointed out in the comments above. 


I have done some research on banded Toeplitz matrices, (where we consider a sequence of finite matrices, and show where the eigenvalues accumulate). There is also quite old literature on this subject, see references in this paper I shamelessly advertise: http://arxiv.org/abs/1208.5607 

