A new version of the Gelfand-Hille theorem

Let $\mathcal{X}$ be a complex Banach space and $A\in \mathcal{L}(\mathcal{X})$ with $\sigma (A)=\{1\}$. We prove that for a vector $x\in \mathcal{X}$, if $\mathrm{\parallel }(A^k+A^{-k})x\mathrm{\parallel }=O(k^N)$ as $k\to +\mathrm{\infty }$ for some positive integer $N$, then $(A-\mathbf{I}{)}^{N+1}x=0$ when $N$ is even and $(A-\mathbf{I}{)}^{N+2}x=0$ when $N$ is odd. This could be seemed as a new version of the Gelfand-Hille theorem. As a corollary, we also obtain that for a quasinilpotent operator $Q\in \mathcal{L}(\mathcal{X})$ and a vector $x\in \mathcal{X}$, if $\mathrm{\parallel }\cos (kQ)x\mathrm{\parallel }=O(k^N)$ as $k\to +\mathrm{\infty }$ for some positive integer $N$, then $Q^{N+1}x=0$ when $N$ is even and $Q^{N+2}x=0$ when $N$ is odd.