Let $T$ be a linear operator of a finite-dimensional vector space $V$ over the field $F$. Let's assume that $T$ has an eigenvalue $\lambda$. Let $K_\lambda$ be the generalized eigenspace of $T$ associated to $\lambda$ and let $W$ be the complement of $K_\lambda$ in $V$ obtained by extending any basis of $K_\lambda$ to a basis of $V$, that is, $V=K_\lambda\oplus W$.
Is $W$ $T$-invariant?