Closed Operators: Spectrum

Question

Given a Hilbert space $\mathcal{H}$.

Consider operators: $$T:\mathcal{D}(T)\to\mathcal{H}$$

Suppose one has: $$T=\overline{T}=T^{**}$$

Then it may happen: $$\sigma(T)=\varnothing,\mathbb{C}$$

What are examples?

Answer 1

Let $\mathcal{H}=L^{2}[0,2\pi]$ and define $L=\frac{1}{i}\frac{d}{dt}$ on the domain of absolutely continuous $f \in \mathcal{H}$ with $f' \in \mathcal{H}$ and $f(0)=f(2\pi)=0$.

This operator is $L$ closed, densely-defined, symmetric and has $\sigma(L)=\mathbb{C}$.

It should be noted that adding the constant function $1$ to the domain of $L$ results in a selfadjoint operator.

An operator with no spectrum: Consider $L=\frac{1}{i}\frac{d}{dt}$ on the domain of absolutely continuous functions $f \in \mathcal{H}$ for which $f' \in \mathcal{H}$ and $f(0)=0$. To see that this has empty spectrum, note that the following equation has a unique solution $f$ for all complex $\lambda$ and $g \in \mathcal{H}$: $$ \frac{1}{i}f'-\lambda f = g,\;\;\; f(0)=0. $$