Invariant subspace of an operator

Let $T:E\to E$ be a linear operator. A linear subspace $M\subseteq E$ is invariant under $T$ if

The invariant subspace is nontrivial when $M\ne\{0\}$ and $M\ne E$. In functional analysis one usually also requires $M$ to be a closed linear subspace.

If $v$ is an eigenvector of $T$, then its one-dimensional span is invariant. Invariance also means that $T$ restricts to a well-defined operator $T|_M:M\to M$.