I want to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e.
$$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$
Some definitions and results of the lecture: (On a Hilbert space $X$ and let $T \in \mathscr{L}(X)$ i.e. a bounded linear operator on $X$)
- $T=T^* \Leftrightarrow \sigma(T) \subset \mathbb{R} $
- $TT^* = T^* T \Rightarrow \sigma_r(T)=\emptyset$ i.e. the residual spectrum is empty
- $\sigma(T)=\sigma_r(T) \cup \sigma_p \cup \sigma_c(T)$ disjoint unions
- If the space is finite then $\sigma_p(T)=\sigma(T)$
- $\sigma(T)$ is non-empty
So, I have a self-adjoint operator i.e the residual spectrum is empty and I also want the point spectrum to be empty i.e. I want to achieve $\sigma(T)=\sigma_c(T)$ i.e
$$ \{\lambda \in \mathbb{C} ~|~ (\lambda I – T) ~\text{not invertible} \}=$$ $$\{\lambda \in \mathbb{C} ~|~ \ker(\lambda I – T)=\emptyset ~\text{ and }~ \text{ran}(\lambda I – T) \neq \overline{\text{ran}(\lambda I – T)}=X \}$$
Additionally the space has to be infinite since otherwise $\sigma(T)=\sigma_p(T)$.
Does someone have such an example for me? And please explain why this example works in this way. This spectral theory is new for me.
Best Answer
Consider the space $H = L^2([0,1],\mu)$, where $\mu$ is the Lebesgue measure. Define $T$ to be the multiplication with the identity function, i.e.
$$(Tf)(x) = x\cdot f(x).$$
Since the identity function is bounded, $T$ is bounded ($\lVert T\rVert \leqslant 1$), and since it is real-valued, $T$ is self-adjoint.
Clearly $T$ has no eigenvalues, since
$$(T - \lambda I)f = 0 \iff (x-\lambda)\cdot f(x) = 0 \quad \text{a.e.},$$
and since $x-\lambda$ is nonzero for all but at most one $x\in [0,1]$, it follows that $f(x) = 0$ for almost all $x$.
We have $\sigma(T) = [0,1]$, as is easily seen - for $\lambda \in \mathbb{C}\setminus [0,1]$, the function $x \mapsto \frac{1}{x-\lambda}$ is bounded on $[0,1]$.
The reason that $T$ has no eigenvalues is that for all $\lambda\in \mathbb{C}$ the set $\operatorname{id}_{[0,1]}^{-1}(\lambda)$ has measure $0$. If $m \colon [0,1] \to \mathbb{R}$ is a bounded measurable function such that $\mu(m^{-1}(\lambda)) > 0$ for some $\lambda$, then $\lambda$ is an eigenvalue of $T_m \colon f \mapsto m\cdot f$, any function $f\in H$ that vanishes outside $m^{-1}(\lambda)$ is an eigenfunction of $T_m$ then. Conversely, if $T_m$ has an eigenvalue $\lambda$ and $f \neq 0$ is an eigenfunction to that eigenvalue, then, since $(m(x) - \lambda)f(x) = 0$ almost everywhere, it follows that $f$ vanishes almost everywhere outside $m^{-1}(\lambda)$, and since $f\neq 0$, it further follows that $\mu(m^{-1}(\lambda)) > 0$.