Let $A$ be a linear operator on a Banach space $X$. Then a residual spectrum $\sigma_\rho (A)$ comprises a bounded or unbounded set of eigenvalues $(\lambda_1,\cdots,\lambda_n,(\cdots))$ on the linear metric space $X$ such that the resolvent, $R_\lambda=(A-\lambda I)^{-1}$ exists but is not defined on a dense set in the space $X$.
By this definition, what is the best example of such a spectrum?
My attempt:
On the Hilbert sequence space $X=l_2$ we define a linear operator $A:l_2\longrightarrow l_2$ by $$(\xi_1,\xi_2,\cdots)\longrightarrow (0,\xi_1,\xi_2,\cdots)$$
where $x=(\xi_j)\in l_2$
Here the resolvent exists $R_0(A)=A^{-1}$. But $0$ is not an eigenvalue, so can we by that assume that the space $(0,\xi_1,\xi_2,\cdots)$ is not dense since it contains a point which is not an eigenvalue? Then the spectrum is not dense, so it is a residual spectrum.