Can anyone help me to understand the definitions of the continuous and essential spectra in simple terms and point out the difference on examples where the two definitions coincide and where don't? How to understand that "Essential spectrum, spectrum of an operator modulo compact perturbations." (wiki)
Further, an intuitive explanation of the difference between the point and discrete spectrum would be very helpful.
Also, how to see the equivalence of the two following definitions of the point spectrum?
$\sigma_p(H) :=\{\lambda \in \mathbb{R} | H −\lambda \text{ is not injective} \}$
and
$\sigma_p(H) :=\{\lambda\in \mathbb{R}| \exists \psi\subset Dom(H), ||\psi||=1, H\psi=\lambda\psi\}$,
how does the "non-injective" relates to the second definition?
Thanks!