# Operators and power regularity

Let $E$ be an infinite-dimensional complex Hilbert space.

An operator $A\in \mathcal{L}(E)$ is called power regular if for all $x\in E$, $r(x,A):=\lim_{n\rightarrow \infty} \|A^nx\|^{1/n}$ exits.

Assume that for all $x,y\in E$, $r(x,A),r(y,A)$ and $r(x+y,A)$ exists.

I want to show that $$r(x+y,A)\leq \max\{r(x,A),r(y,A)\} .$$

Letting $n\to\infty$ gives