I'm following Kreyszig's "Introductory Functional Analysis with Applications" and am trying to follow the proof of the following Theorem (9.2-4 on p. 468)
For a bounded self-adjoint linear operator $T:H\to H$ on a complex Hilbert space $H$, $$\sigma_{r}\left(T\right)=\emptyset,$$
i.e. its residual spectrum is empty.
The proof refers to the following Lemma:
Lemma (projection theorem) Suppose that $Y$
is a closed subspace of a Hilbert space $H$. Then
$$
H=Y\oplus Y^{\perp}.
$$
Kreysig's begins his argument as follows.
"Suppose, for contradiction, that $\sigma_{r}\left(T\right)$ is non-empty;
indeed, pick $\lambda\in\sigma_{r}\left(T\right)$. Then by definition of the residual spectrum, $R_{\lambda}:=T_{\lambda}^{-1}=(T-\lambda I)^{-1}$
exists but its domain is not dense in $H$."
I understand everything so far. He then goes to argue:
"Hence by the Projection Theorem, there is a $y \neq 0$ in $H$ which is orthogonal to the domain of $R_{\lambda}$."
I don't see why this is. I wonder if someone could very kindly explain?
Best Answer
Denote $Y=\operatorname{Im}(T_\lambda)$, then $Y$ is a proper subspace of $H$. By projection theorem $Y^\perp\neq \{0\}$, so you can choose $y\in Y^\perp\setminus\{0\}$ which is by definition of $Y$ is orthogonal to $Y$, i.e. to $\operatorname{Im}(T_\lambda)$