In this article http://en.wikipedia.org/wiki/Decomposition_of_spectrum_(functional_analysis) spectrum decomposition of the shift operator on $\ell_p(N)$ has been discussed.

**Question**:
Is it possible figure out the decomposition **without help of the adjoint operator**?

For example, to show that the open unit disc, say, {$\lambda\in{\bf C}:|\lambda|<1$} includes in the residual spectrum of the right shift operator $R$, one needs to show that range$(R-\lambda I)$ is not dense in $\ell_p(N)$ or equivalently, there is an $x\in \ell_p(N)$ such that it is not adherent to range$(R-\lambda I)$.

Functional Analysis(Chapter 4 I think?) – Yemon Choi Mar 17 '11 at 19:03