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?