Let $A$ be a fixed real matrix such that $A$ can be written as $A = B + C$ for some $B, C$. Suppose that for any polynomial $f$ we have $f(A) = f(B) + f(C)$.
When is it true that $g(A) = g(B) + g(C)$ when $g$ is not a polynomial but an analytic function?
Writing $g = \lim_{n \to \infty} g_n$ with $g_n(x) = \sum_{k=0}^n \alpha_k x^k$, I have that $g_n(A) = g_n(B) + g_n(C)$ for each $n$. Is this enough to ascertain the same equality for $g$? Why or why not?
Any help appreciated!
This is true. If $g$ is analytic on $\mathbb C$ the it has an expanseion $g(z)=\sum\limits_{n=1}^{\infty} a_nz^{n}$ valid for all $z$. This implies that $\sum\limits_{n=1}^{\infty} |a_n| R^{n}<\infty$ for all $R>0$. It follows that the series $\sum\limits_{n=1}^{\infty} a_n M^{n}$ converges in the matrix norm for any $M$. So applying the hypothesis for the partial sums of $\sum\limits_{n=1}^{\infty} a_nz^{n}$ and taking limits yields the conclusion. [Note that $g(A), g(B),g(C)$ are defined by taking limits of the partial sums of the corresponding series].