I have asked this question previously at Math.stackexchange, but it seems to receive little attention there. 


In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer science), I have come upon the necessity of solving (or at least finding out some of the solution's properties) infinite systems of difference (NOT differential) equations (linear, autonomous, first-order). Solving countable systems is sufficient for my purpose.


However, I have almost no idea about how to solve such systems (and what are the solvability conditions). I have a feeling that a possible approach is to seek generalizations of the finite-dimensional method to infinite-dimensional spaces via the spectral theory of linear operators, but I have no idea about the details.


So my question is whether there are any resources that deal systematically with infinite systems of difference equations. I would be grateful either for a resource that can be read also without a knowledge of advanced mathematics (that is something readable for non-mathematicians) or, alternatively, for a more advanced resource with a recommendation where to learn the prerequisities. My mathematical background is unfortunately quite basic (I am a computer science major): possibly relevant fields I have a background in are calculus, linear algebra, some fundamentals of modern analysis, difference/differential equations, and basics of functional analysis (however I have very poor knowledge of spectral theory).


I would appreciate any resources dealing with this issue, as well as any recommendations about which parts of mathematics I am supposed to learn (preferably with some recommended resources). I am willing to invest a considerable amount of time into the study, but I would like to follow some recommended path that leads to the goal. In the case that the theory of infinite systems of differential equations is similar, I would appreciate also resources dealing with them.



Resource on Infinite Systems of Difference Equations
If your equations look like $x\_{i+1} = Ax\_i$, where each $x\_j$ is an element of a (possibly infinite dimensional) vector space and $A$ is a linear operator thereon, then the general solution is $x\_j = A^j x\_0 = \exp(j\log(A))x\_0$, with $x\_0$ specifying the initial conditions. The properties of $A^j$, or equivalently $\exp(j\log(A))$, can be deduced from a knowledge of the spectrum and invariant subspaces of $A$. This is precisely spectral theory. So I guess you just have to bite the bullet and start reading up on spectral theory.


The easiest situation is when the $x\_j$ are vectors in a Hilbert space and $A$ is a bounded self-adjoint operator. This case is covered in essentially every introductory book on functional analysis. There is a long list of references of introductory and higher level reference books on [Wikipedia](http://en.wikipedia.org/wiki/Functional_analysis).


If your $A$ is not of the above form, then you need to be more specific about the structure of your difference equation to get a more relevant suggestion.


