# Compact Integral Operators induced by positive Kernels

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.