Let $\Omega \subset \mathbb{C}$ open and bounded, $X = C^0(\overline{\Omega})$ (vector space of continuous complex-valued functions). For an $y \in X$ with $y(\overline{\Omega}) = \{\lambda \in \mathbb{C} : \vert \lambda \vert \leq 1\}$ let $Tx = y \cdot x$. Calculate the spectra $\sigma_r$, $\sigma_p$ and $\sigma_c$.
From $\Vert Tx \Vert = \Vert y \cdot x \Vert \leq \Vert x \Vert$ it follows that $\Vert T \Vert = 1$, so I know that $\sigma(T) \subseteq \overline{B_1(0)}$. From $((\lambda - T)x)(t) = (\lambda - y(t))x(t)$ it follows that $\sigma_p = \emptyset$. I guess I have to divide this into different cases now and solve them one after another?
Is this true so far? How do I continue? I'm quite new to spectra so please, don't be that hard on me. (English also ain't my first language, sorry again)
Edit: Okay thanks alot user108903!