Spectrum of an unbounded operator $A$ Are there any reference that we can find the following
Theorem:  Let $A$ be a positive (resp. negative) self-adjoint unbounded operator on a Hilbert space $H$. Then, the spectrum of $A$ is contained in
the positive real-axis (resp. negative real-axis).
 A: Assume that $A : \mathcal{D}(A) \subset H\rightarrow H$ is an unbounded selfadjoint operator on a real or complex Hilbert space $H$ such that $(Ax,x) \ge 0$ for all $x\in\mathcal{D}(A)$. For $\lambda > 0$,
$$
         \lambda(x,x) \le ((A+\lambda I)x,x)\le \|(A+\lambda I)x\|\|x\|,\;\;\; x\in\mathcal{D}(A).
$$
Therefore,
$$
                 \lambda\|x\| \le \|(A+\lambda I)x\|,\;\; x\in\mathcal{D}(A).
$$
Such an inequality forces the range of $A+\lambda I$ to be closed because because, if $(A+\lambda I)x_n$ converges to $y$, then $\{ (A+\lambda I)x_n \}$ is a Cauchy sequence and the above forces $\{ x_n \}$ to be a Cauchy sequence, which must converge to some $x$ because $H$ is complete. Then, because $A$ is closed, $y = (A+\lambda I)x$. So the range of $A+\lambda I$ is closed. The range of $A+\lambda I$ must be all of $X$ because it is closed and $y \in \mathcal{R}(A+\lambda I)^{\perp}$ iff
$$
             ((A+\lambda I)x,y)=0,\;\;\; x\in\mathcal{D}(A).
$$
By the definition of adjoint, $y \in \mathcal{D}(A^*)=\mathcal{D}(A)$ and $(A+\lambda I)y=0$. However, $\mathcal{N}(A+\lambda I)=\{0\}$ which gives $y=0$. Hence, $-\lambda\in\rho(A)$, as was to be shown.
