Spectral decomposition and Gelfand's theorem

Abstract

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 $σ(A)=\{iλ_k; k\in\mathbb{Z}^* \}$ is discrete and satisfies $\sum \frac{1}{|λ_k|^\ellδ_k^n}<\infty$ ($n$ 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 $δ_k=\min(\frac{| λ_{k+1}-λ_k|}2, \frac{|λ_{k-1}-λ_k|}2)$.