Why is this operator self-adoint

Question

We have that $\lambda, \overline{\lambda} \in \rho(T)$ and $\lambda \in \mathbb{C}$. Now, I want to show that a symmetric operator and closed operator $T: \operatorname{dom(T)} \rightarrow H$ must be self-adjoint. Notice, that $T$ is not necessarily densily defined. Does anybody here have any ideas? Actually, I concluded the closedness of this operator from the fact that the resolvent is not empty by myself, so this may be somehow a tautology in this exercise.

Answer 1

If $A$ is symmetric and $(A-\lambda I)$ is surjective for for some $\lambda\notin\mathbb{R}$, then the domain of $A$ is automatically dense. To see this, assume $A-\lambda I$ is surjective for some non-real $\lambda$ and suppose that $y \perp \mathcal{D}(A)$. Then $y=(A-\lambda I)x$ for some $x \in \mathcal{D}(A)$, which gives $$ 0=(y,x)=((A-\lambda I)x,x)=((A-\Re\lambda I)x,x)+i\Im\lambda(x,x). $$ Because $A$ is symmetric, then $(Ax,x)=(x,Ax)$ must be real. Therefore, $$ 0 = \Im ((A-\lambda I)x,x) = \Im\lambda\|x\|^{2} \implies x = 0\implies y = 0. $$ The conclusion is that $A$ must be densely-defined if $A$ is symmetric and if $A-\lambda I$ is surjective for some $\lambda \notin \mathcal{R}$. Therefore $A^{\star}$ is closed and densely-defined with $A\preceq A^{\star}$, meaning that $A$ is a restriction of $A^{\star}$ to $\mathcal{D}(A)$.

Now assume $A$ a densely-defined symmetric operator. Then $A$ is closable, which implies that $A^{\star}$ is closed and densely-defined, even if $A$ is not closed. Assuming $A-\lambda I$ and $A-\overline{\lambda}I$ are surjective for some $\lambda$, I'll now show that $A=A^{\star}$ (here $\lambda$ can be real.) We know that $A\preceq A^{\star}$ because $A$ is symmetric; so it is enough to show that $y \in \mathcal{D}(A^{\star})$ implies that $y \in \mathcal{D}(A)$. To prove this, assume $y\in\mathcal{D}(A^{\star})$ and choose $z \in \mathcal{D}(A)$ such that $$ (A-\overline{\lambda}I)z = (A^{\star}-\overline{\lambda})y. $$ Then, for all $x\in\mathcal{D}(A)$ one has $$ ((A-\lambda I)x,y)=(x,(A^{\star}-\overline{\lambda}I)y)=(x,(A-\overline{\lambda})z)=((A-\lambda I)x,z),\\ ((A-\lambda I)x,y-z) = 0,\;\;\; x \in \mathcal{D}(A). $$ Because $A-\lambda I$ is also surjective, then $y = z \in \mathcal{D}(A)$, as was to be shown.

Added: If you assume that $A$ is densely-defined and symmetric with $A-\lambda I$ surjective for some real $\lambda$, then $A=A^{\star}$. That's because the arguments of the second paragraph above remain valid. Alternatively, if $A-\lambda I$ is surjective for some $\lambda$ for which $((A-\lambda)x,x) > 0$ for all non-zero $x \in\mathcal{D}(A)$, then $A=A^{\star}$ is a closed densely-defined selfadjoint operator.