Define the integral operator $K: L^2(0,1) \to L^2 (0,1)$ by: $$ Kf(x) = \int_0^1 e^{x-y} f(y) \mathrm{d}y $$
- Compute the range of $K$. Is it a closed set? Is the operator compact?
- Compute the adjoint $K^{*}$ and find its kernel.
- Verify that the equation $Kf = g$ if and only if $f \perp \ker K^*$
I am having issues with the first part of the problem. Namely, I am not sure how it is possible to characterize the range of this operator (in a tractable way).
As for the second part, I think using Fubini's theorem and rearranging the integrals should work.
Finally, I am not sure how to prove the last part. I know that $L^2 = \overline{\mathrm{ran}(A)} \oplus (\ker(A*)$ for any bounded operator $A$, but I am not sure how to apply this information.