How to represent non-linear operators computationally?

Question

I have a finite dimensional vector space V, and want to compute a non-linear operator $R: V \rightarrow V$.

I want to have a "general" form of this operator R. I think of the following series expansion: $$ R = a_0 I + a_1 T + a_2 T^2 + a_3 T^3 + ... $$ where T is a linear operator over V. But it seems that the above R would become a matrix and that is itself a linear operator. Something wrong?

Secondly, if the above series is correct, and if I truncate the series to approximate the operator, I am not sure if the truncation has the "nice" property as in truncation of the series expansion of a real or complex single-variable function...

I just want a general form of a non-linear operator so my algorithm can approximate it. My training data takes the form of some vectors in V and after the transformation they would become some other vectors.

Edit: I assume people working in numerical methods, would have some ways to manipulate non-linear operators. What kind of representations are available for them? (Matrices are linear and their polynomials and power series are still linear.)

Answer 1

Maybe you are interested in the Functional Calculus ( of different types) https://en.wikipedia.org/wiki/Functional_calculus ?

Answer 2

First part: Assuming the series on right does converge to R, then R will be linear operator. For ex. For any matrix $A$, $$exp(A) := \sum_{n=0} ^{\infty} \frac{A^n} {n! } $$ is a linear operator as $exp(A)(k u + t v) =k ~exp(A)u +t~ exp(A)v $ for real number $k, t$. Regarding second part, series will have all the "nice" properties like series of real or complex numbers if it is known to be convergent (in your appropriate space).