I want to find an example where the Spectrum is equal to the Continuous Spectrum in C[0,1]; $\sigma_c(A)=\sigma(A)$

Question

I tried to find this example but the condition $\overline{\operatorname{range}(\lambda I -A)}=C[0,1]$ is too hard to prove. Anyone could help me?

Answer 1

Consider the Hilbert space $H=L^2((0,1))$ and let $A:D(A) \to H$ be defined by

$(Af)(x)=xf(x),$ where $D(A)=\{f \in H: xf \in H\}.$

Then $A$ is self-adjoint, hence the residual spectrum of $A$ is empty. Furthermore, $A$ has no eigenvalues.

Conclusion:

$$\sigma(A)= \sigma_c(A)=[0,1].$$