What are the necessary and sufficient conditions to allow for the same manipulation of non-polynomial mathematical entities like matrices, linear transformations, differential operators, etc. as for polynomials?
For instance, if $T\in \mathcal{L} (V)$, where $V$ is a finite-dimensional vector space over $\Bbb{C}$, then for any $a_0,\ldots, a_m \in \Bbb{C}$, we can factor $$a_0I+a_1T+\cdots +a_mT^m\in \mathcal{L} (V)$$ as $$c(T-\lambda_1)\cdots (T-\lambda_m)$$ where $c=a_0$ and $\lambda_1,\ldots, \lambda_m \in \Bbb{C}$.
I’m mainly interested in the complex field where I can factor any polynomial entirely.