What can be said about the sequence $(\left\Vert A^{n}\left( x\right) \right\Vert ^{\frac{1}{n}})_{n}$

Question

Let $H$ be a separable infinite dimensional Hilbert space and let $A\in B(H)$ be a bounded operator. For any $x\in H$ is $(\left\Vert A^{n}\left( x\right) \right\Vert ^{\frac{1}{n}})_{n}$ a decreasing and/or a convergent sequence ? Recall that $\lim \left\Vert A^{n} \right\Vert ^{\frac{1}{n}}$ is the spectral radius of $A$.