0
$\begingroup$

Let $K$ be a compact operator induced by the kernel $k(s,t)\in L^2([0,1])^2$ with $k(s,t)>0$. Prove that $\|K\|<1$ if and only if $(I-K)$ has a bounded inverse $(I-K)^{-1}$ which is induced by a positive kernel.

I am mainly confused how $(I-K)^{-1}$ is induced by a kernel at all. If we assume $\|K\|<1$, then $(I-K)^{-1}=\sum_{j=0}^\infty K^j$ is indeed bounded, but why is it induced by a kernel? I could maybe show that $\sum_{j=1}^\infty K^j$ is induced by a positive kernel, but is $I$ itself even an integral operator, i.e. induced by a kernel? Am I misunderstanding something? I would be really happy if someone could explain to me how this is meant.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.