  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 $\|(A^{k}+A^{-k})x\|=O(k^N)$ as $k \rightarrow
+\infty$ 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 $\|\cos(kQ)x\|=O(k^N)$ as $k \rightarrow +\infty$ 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.
