Consider the vector spaces $D_1$, $D_2$, $D$ and $X$ such that $D\subset X$ and $D=D_1\oplus D_2$. Furthermore, suppose that $L:X\longrightarrow D$ is a linear map such that $D_1$ is stable under $L$, that is, if $x\in D_1$, then $Lx\in D_1$.
My question is: can we prove the stability of $D_2$ under L? If not, can we add conditions on $L$ to achieve this result?
No; as is typical in the subject this is already false for $2 \times 2$ matrices. Take $D = X = \mathbb{R}^2, D_1 = \text{span}(e_1), D_2 = \text{span}(e_2)$. Then it suffices to take $L : \mathbb{R}^2 \to \mathbb{R}^2$ to be a linear map which has $e_1$ as an eigenvector but not $e_2$, for example
$$L = \begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$$
where $a, b, c \neq 0$.
A sufficient condition is that $D = X$ is a Hilbert space, $L$ is normal (for example, self-adjoint, skew-adjoint, or unitary), and $D_2 = D_1^{\perp}$ is the orthogonal complement.
