Hello I'm wondering about how to deal with a situation like this:
I have $\mathcal{H} = L^2(0,\pi /2)$ and
\begin{equation} Tf(x) = \int_0^{\pi/2} f(y)\cos(y)\sin(x)\,dy \end{equation}
First I have to show that $T$ belongs to the space of finite rank operators , but this is trivial cause I can identify Im$(T)$ as span of $h(x) := \sin(x)|_{(0,\pi/2)}$ (right?). Then I have to determine its spectrum. So I know that \begin{equation} \lambda \in \sigma_p (T) \Leftrightarrow\exists f\in L^2(0,\pi /2), f \neq 0 \end{equation}
such that $Tf = \lambda f$. In this case I have
\begin{equation} \sin(x)\int_0^{\pi/2} f(y)\cos(y)\,dy = \lambda f(x) \end{equation}
Should I now derive both sides of equation, assuming $f \in C^1 (0,\pi/2)$?