# condition for postive symmetric operator to be essentially self-adjoint

Suppose $$A$$ is a densely defined symmetric operator on Hilbert space which is positive. (a) prove $$||(A+I)\phi||^2\geq ||\phi||^2+||A\phi||^2$$ (b) Show $$Ran(A+I)$$ is closed if $$A$$ is a closed operator (c)Prove that A is essentially self-adjoint iff $$A^*\psi=-\psi$$ has no non-zero solution.

I got stuck on part (c). We know $$(A\pm iI)$$ has dense image iff $$A$$ is essentially image. Also, I am thinking about applying part (b) result to $$A^*+I$$. Anyone could possibly give me a hint?

Theorem: Let $$A : \mathcal{D}(A)\subseteq\mathcal{H}\rightarrow\mathcal{H}$$ be a densely-defined symmetric linear operator on a Hilbert space $$\mathcal{H}$$, and suppose that $$\langle Ax,x\rangle \ge 0$$ for all $$x\in\mathcal{H}$$. Then $$A$$ is essentially self-adjoint iff the range of $$A+I$$ is dense in $$\mathcal{H}$$.
The range of $$A+I$$ is not dense iff $$\langle (A+I)x,y\rangle=0$$ for some $$y\ne 0$$, which is equivalent to the existence of $$y\in\mathcal{D}(A^*)\setminus\{0\}$$ and $$(A^*+I)y=0$$.