Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{R} \to B(\mathscr{H}), r \mapsto S_r$ be a continuous (or smooth) family of operators, where $B(\mathscr{H})$ is the space of bounded operators on $\mathscr{H}$.

Denote by $\rho(S_r)$ the spectral radius of $S_r$ and assume that $(\sup_{r \in [0,1]}\rho(S_r)) < 1$. I want to show for large enough $n$, that $$\left(\sup_{r \in [0,1]} ||S_r^n||^{\frac{1}{n}} \right) < 1.$$ Does this hold in this generality? If no, what would one need to assume in addition in order for this to hold? Clearly this would hold if the operator $S_r$ are all normaloid (i.e. when $\rho(S_r) = ||S_r||$), yet I don't want to assume this.

Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{R} \to B(\mathscr{H}), r \mapsto S_r$ be a continuous (or smooth) family of operators, where $B(\mathscr{H})$ is the space of bounded operators on $\mathscr{H}$.

Denote by $\rho(S_r)$ the spectral radius of $S_r$ and assume that $(\sup_{r \in [0,1]}\rho(S_r)) < 1$. I want to show for large enough $n$, that $$\left(\sup_{r \in [0,1]} ||S_r^n||^{\frac{1}{n}} \right) < 1.$$ Does this hold with the above assumptions? If not, what would one need to assume in addition in order for this to hold? Clearly this would hold if the operator $S_r$ are all normaloid (i.e. when $\rho(S_r) = ||S_r||$), yet I don't want to assume this.