Operator norm in Hilbert space

Question

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $F$.

If $A\in \mathcal{B}(F)$, why $(\|A^n\|^{1/n})_n$ is a decreasing sequence?

Answer 1

The claim is false. Let $A=\begin{bmatrix}0&2\\1/2&0\end{bmatrix}.$ Then $A^2 = I$, $A^3 = A$, $A^4 = I$, and so on. The sequence $(\|A^n\|^{1/n})_n$ is $(2,1,2^{1/3},1,2^{1/5},1,\dots)$.