I have seen two different kinds of definitions on spectrum, one is from Wiki and another one can also be found on many materials.
Suppose that $A$ is a bounded linear operator on a Hilbert space $\mathcal{H}$.
1st version:
the set-theoretic inverse is either unbounded or defined on a non-dense subset[Kreyszig, Erwin. Introductory Functional Analysis with Applications.]
2nd version:
(a)point spectrum(eigenvalue): $\sigma_p:=\{\lambda\in\sigma(A)| A-\lambda I \text{ is not one to one} \}$
(b)continuous spectrum: $\sigma_p:=\{\lambda\in\sigma(A)| A-\lambda I \text{ is one to one but not onto, and $ran(A-\lambda I)$ is dense in $\mathcal{H}$} \}$
(c)residual spectrum: $\sigma_p:=\{\lambda\in\sigma(A)| A-\lambda I \text{ is one to one but not onto, and $ran(A-\lambda I)$ is not dense in $\mathcal{H}$} \}$.
So, are they equivalent? If so, how shall I prove these two definitions are actually equivalent? Will the Close Graph Theorem be used?
