Eigenvalues of a Hilbert-Schmidt operator

Question

Let $\alpha>1/2$. What is the point spectrum of the operator $Tf(x)=\int_0^1(x^\alpha+y^\alpha)f(y)dy$ on $L^2([0,1],dx)$? Nobody ever showed me how to calculate the eigenvalues for this kind of operators.

Answer 1

Ok, I managed to solve this. It's just a matter of writing, for some $\lambda>0$ candidate and $f\neq0$ in $L^2$, $\lambda f(x)=x^\alpha\int_0^1 fdy+\int_0^1y^\alpha fdy$, so that $f(x)=Ax^\alpha+B$ for some $A,B\in\mathbb{C}$. Hence, $\int_0^1(Ax^\alpha+b)(x^\alpha+y^\alpha)dy=\lambda(Ax^\alpha+B)$. The left hand side gives a polynomial in $x$ that must be equal a.e. to the right hand side. Thus we have to solve a linear system $2\times 2$ and this is easy to do. Then, one has to verify that $A,B$ found satisfy the eigenvalue equation and for which $\lambda$ they do...