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 $\sigma (A)=\{i{\lambda }_k;k\in {\mathbb{Z}}^{\ast }\}$ is discrete and satisfies $\sum \frac{1}{|{\lambda }_k{|}^{\ell }{\delta }_k^n}<\infty$ ($n$ and $\ell$ nonnegative integers), then there exists projectors $(P_k{)}_{k\in {\mathbb{Z}}^{\ast }}$ such that $\sum P_{k}x=x$ ($x\in D(A^{n+\ell })$), where ${\delta }_k=\min (\frac{|{\lambda }_{k+1}-{\lambda }_k|}{2},\frac{|{\lambda }_{k-1}-{\lambda }_k|}{2})$.

15 pages, 1 figure