I want to find the spectrum $\sigma(T)$ of the following operator: $$(Tx)(s)=\int_0^1 s(s^2+2) x(t) \ dt.$$ Where $x\in C[0,1].$
Since $T$ is compact and self-adjoint, the spectrum of $T$ consists of zero and real eigenvalues. So, I assume that $\lambda x(s)=\int_0^1s(s^2+2) x(t) \ dt.$ But I don't know how to continue. Could you help me? Thank you.
$\lambda x(s)=s(s^{2}+2) \int_0^{1} x(t)dt$. Integrate w.r.t. $s$ You get $\lambda \int_0^{1} x(s)ds=\frac 5 4 \int_0^{1} x(t)dt$. If $\int_0^{1} x(t)dt \neq 0$ this gives $\lambda =\frac 5 4$. If $\int_0^{1} x(t)dt=0$ we get $\lambda x(s)=0$. Since $x \neq 0$ we get $\lambda =0$.
Can you check that $T$ is not surjective? That would tell you that $T$ is not invertible, so $0$ is certainly an eigen value. [Alternatively any non-zero function with $\int_0^{1} x(t)dt=0$ is an eigen function corresponding to eigen value $0$]. $\frac 5 4 $ is an eigen value with eigen function $t(t^{2}+2)$. Hnece the spectrum is $\{0,\frac 5 4 \}$.