Let $X$ be a real normed space and $S, T$ bounded operators on $X$.

• $\rho_0(T) = 0$, $\rho_1(T) = \|T\|$, $0 \le \rho_k(T) \le \|T\|$, and $\rho_q(T) \le \rho_p(T)$ for $1 \le p \le q$.
• $g(T, V) \le \rho_m(T)$ for every $V \in \mathcal{G}_m X$, and $\rho_1(T) \cdots \rho_k(T) \le \|T\|^k$.

For every $k$,

Moreover, for $k \ge 1$ the growth $g(T, V)$ is jointly continuous in $(T, V)$, for the operator norm on $T$ and the Hausdorff distance between unit spheres on $V \in \mathcal{G}_k X$.