Example: Operator with empty spectrum

Question

I tried Google and a few books but couldn't find a suitable example. Does anyone know an example of an (unbounded closed) Operator BETWEEN HILBERTSPACES(!), that has empty spectrum?

Thanks for your help!

Answer 1

A closed densely-defined linear operator on a Hilbert space can have empty spectrum. For example, let $H=L^{2}[0,1]$ and let $A=\frac{d}{dx}$ on the domain consisting of absolutely continuous $f \in L^{2}$ for which $f(0)=0$ and $f' \in L^{2}$. To show that $A$ has no spectrum, it is enough to prove that the resolvent $R(\lambda)=(A-\lambda I)^{-1}$ exists everywhere. To do this, given $g\in L^{2}$, solve $$ f'-\lambda f = g, \;\;\; f(0)=0. $$ The unique solution is trivially obtained: $$ \frac{d}{dx}(e^{-\lambda x}f)=e^{-\lambda x}g \\ f(x)=e^{\lambda x}\int_{0}^{x}e^{-\lambda t}g(t)dt $$ It is is easy to verify that $f$ is the correct solution. So $A$ has empty spectrum because its resolvent exists everywhere: $$ R(\lambda)g = (A-\lambda I)^{-1}g = \int_{0}^{x}e^{\lambda(x-t)}g(t)dt. $$