Consider the Banach space $X=C[0,1]$ of continuous functions $f:[0,1]\to\mathbb{R}$ equipped with the supremum norm. If we consider the following unbounded operator $A$ defined on its domain $D(A)=\{f\in C^1[0,1]:f(0)=1\}$ by $Af=f'$. I am interested in finding its spectrum. So if $g\in X$, then $g$ has a unique antecedent $f$ by the application $\lambda-A$ which is given by the formula $$f(x)=e^{\lambda x}-\int_0^xe^{\lambda(x-y)}g(y)dy.$$ That is $\lambda-A$ is invertible for every $\lambda\in \mathbb{C}$. Which means that the spectrum of $A$ is empty.
Now in the other hand, for each $\lambda\in \mathbb{C}$, if we consider the function $f_\lambda$ defined by $f_\lambda(x)=e^{\lambda x}$, then $f_\lambda\neq0$, $f_\lambda\in D(A)$ and $Af_\lambda=\lambda f_\lambda$, which means that the point spectrum is the whole complex plane. But this contradicts the fact that the whole spectrum is empty !!?