Spectral decomposition and Gelfand's theorem

In this paper we are interested in spectral decomposition of an unbounded operator with discrete spectrum. We show that if $A$ generates a polynomially bounded $n$-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_k; k\in\mathbb{Z}^* \}$ is discrete and satisfies $\sum \frac{1}{|\lambda_k|^\ell\delta_k^n}<\infty$ ($n$ and $\ell$ nonnegative integers), then there exists projectors $(P_k)_{k\in\mathbb{Z}^*}$ such that $\sum P_kx=x$ ($x\in D(A^{n+\ell})$), where $\delta_k=\min(\frac{| \lambda_{k+1}-\lambda_k|}2, \frac{|\lambda_{k-1}-\lambda_k|}2)$.