Spectral decomposition and Gelfand’s theorem
AbstractWe consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set 
$\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}$ 
is discrete and satisfies 
$\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty$
, where ℓ is a nonnegative integer and 
$\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2})$
. In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors 
$(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}}$ 
such that, for any x∈D(An+ℓ), the decomposition ∑Pkx=x holds.