Consider a densely defined operator $A : C[a,b ]\rightarrow C[a,b ]$, $$Au=u^{\prime}$$ with domain $$D(A)= \{ u\in C^1[a,b]: u(b)=ku(a) \}$$ for some $k>0$.
I have to find $R_A(\lambda)$ for $k\neq 1$, it looks like the sum of two integrals.
But how to find $R_A(\lambda)$ for $k=1$. I cannot find an idea to present it as integral. Maybe I need to use Neumann series, but I cannot prove that $\|A\|<1$.