Thank you for the extensive comment. This seems like a path to follow. However, I shall try to leave this question open for some time, mainly in order to seek some other reference on the general theory you have mentioned.

@anonymous Thank you for your answer. To be honest, I am not very knowledgeable in this algebraic stuff, so I am not able to figure out, if it is relevant to me or not. However, the number of dependent variables at each time step is in deed infinite. Anyway, this seems like an interesting stuff, but I am afraid that more for experts than for people like me.

I have one more question... By $\exp(A)$ do you denote $e^A$? Because this sounds to me more like a solution to a differential equation than to a difference equation. But I am surely missing something...

Thank you for your answer, this sounds good... My equations are exactly of the form $x_{i+1} = Ax_i$, however $A$ need not be self-adjoint. In my basic setting, $A$ is a bounded operator on $\ell^{\infty}$ that can be viewed as an infinite matrix that is $k$-diagonal for some constant $k$. However, there may be a possibility to transform a problem to a more usual space $\ell^2$.

@ThomasRichard Yes, linear and autonomous.