Determining the different spectra of an operator

Question

So my question is how to exactly determine the different types of spectra (regular / continuous / point / resiudal) of an operator.
To use as an illustrative example, we consider the linear operator $A: l_2(\mathbb{N}) \to l_2(\mathbb{N})$ given by $$ Ax:= \bigg ( \big( \frac{1}{n}+ \frac{4i}{n^2})x_n \big ) \bigg )_{n \in \mathbb{N}} $$ So, I know to find the point spectrum $\sigma_p$, we have to solve $$ (A-\lambda \mathbb{I})x = 0 $$ to find the points where $A-\lambda \mathbb{I}$ is not injective. How to find the points of $\sigma_c$ and $\sigma_r$? I know that these are both cases for $\lambda$ where $A-\lambda \mathbb{I}$ is not surjective but not how to find these.

Answer 1

The operator is of the form $$Ax=\{a_nx_n\}_{n=1}^\infty,\qquad a_n\neq 0,\ a_n\to 0$$ Let $\{e_n\}_{n=1}^\infty $ denote the standard basis in $\ell^2(\mathbb{N}).$ Then $Ae_n=a_ne_n.$ Hence $$\{a_n\,:\, n\in \mathbb{N}\}\subset \sigma_p(A)$$ The operator $A$ is injective and $${\rm Im}\,A=\{y\in \ell^2(\mathbb{N})\,:\, \{y_n/a_n\}\in \ell^2(\mathbb{N})\}\subsetneq \ell^2(\mathbb{N})$$ The space ${\rm Im}\,A$ is dense, as $e_n\in {\rm Im}\,A$ for any $n.$ Hence $0\in\sigma_c(A).$ For $z\notin \{a_n\,:\, n\in \mathbb{N}\}\cup\{0\}$ we have $|z-a_n|\ge \delta$ for a positive number $\delta.$ Then the operator $$B_zx=\{(a_n-z)^{-1}x_n\}_{n=1}^\infty$$ is bounded, namely $\|B_z\|\le \delta^{-1}$ and $B_z(A-zI)=(A-zI)B_z=I.$ Hence $z\notin\sigma(A).$