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 $\delta_k=\min(\frac{ | \lambda_{ k+1}-\lambda_k|}2, \frac{ |\lambda_{ k-1}-\lambda_k|}2)$.
Categories: math.SP