In books on time series the following is frequently used:
We are given a power series $C(z):=\sum_i C_i z^i$, wherein the $C_i$ are matrices and the series is potentially infinite. Subsequently a relation is shown, e.g. a connection between two such series: $$C(z) = C^*(z)(1-z)$$ afterwards the text says that it is possible to insert the lag-operator $L$ for $z$ (basically this operator takes as input a stochastic process and produces a lagged version as output).
So the implication is that we may simply plug-in $L$ for $z$ and derive $$C(L) = C^*(L)(1-L)$$
My question: Can someone point me either to a reference where are precise mathematical justification is given o- or: which type of formal prerequisites do I need to understand this. I am specifically asking this because the overall structural setup that is required is even unclear to me. For instance, I am aware of the algebraic structure of a formal polynomial in an abstract variable $T$ with coefficients from a (commutative) ring. But this does not seem to suffice in this case.
EDIT:
Just for reference: This line of reasoning appears in the analysis of cointegration in vector autoregressive models, e.g. in Johansens book: "Likelihood-Based Inference in Cointegrated Vector Autoregressive Models" -> https://academic.oup.com/book/27916?login=false
It appears for example in the derivation that a I(1) process is non-stationary. (p.36). But there are many other places where this line of reasoning is used.