Consider the space of continuous realvalued functions $C[a,b]$ and an integral kernel $k \in C([a,b]\times[a,b])$, where $ba<1$. The operator $K: L^2[a,b]\to L^2[a,b]$, $$ K f(x) =\int_{a}^{bx+a} f(s) k(s,x) ds, \quad f\in L^2[a,b]$$ is HilbertSchmidt. Prove that the linear operator $K$ has no eigenvalues.
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$\begingroup$ Of course, $0$ can be an eigenvalue. For $\lambda\not= 0$, I think you can observe that an eigenfunction would have to be continuous, then solve by iteration and control the $L^{\infty}$ norm. $\endgroup$ – Christian Remling Jan 28 at 18:28