Ok so the question is very simple, if you have the Hilbert Schmidt operator: $$Kf[x]=\int_a^b k(x,y)f(y)dy,$$ with $f\in L^2(a,b)$, how can you find his eigenvalues(i.e, $Kf_n=\lambda_n f_n$)? You need to solve some integral equations? Thanks
[Math] Eigenvalues of the Hilbert – Schmidt operator
eigenvalues-eigenvectorsfunctional-analysishilbert-spaces
Best Answer
If your problem comes from a more strongest formulation like, for example, this ODE $-u''(x)+\mu u(x)=f(x)$ with some boundary conditions try to use the condition of eigenvalue there.
Continuing with the example, you know that $f \neq 0$ will be an eigenvector with eigenvalue $\lambda$ if and only if $-\lambda f''(x)+\mu \lambda f(x)=\lambda f(x)$. So you are looking for the values of $\lambda$ that make that $-\lambda f''(x)+\mu \lambda f(x)=\lambda f(x)$ has more than one solution. You just need to solve the equation and use the boundary conditions to check if there is any non-trivial solution.