Let $X$ be a real normed space, $k \ge 1$, $T$ a bounded operator on $X$ and $\Theta$ with $\rho_{k+1}(T) < \Theta < \rho_k(T)$. Let $V, W \in \mathcal{G}_k X$ with $g(T, V) > \Theta$ and $g(T, W) > \Theta$. Then $TV, TW \in \mathcal{G}_k X$ and, writing $x = \rho_{k+1}(T)/\Theta \in [0, 1)$,

So an operator with a gap between $\rho_{k+1}$ and $\rho_k$ contracts the subspaces on which it grows fast (Lee 2024, Lemma 12).