A is essentially self-adjoint if and only if the equation $A^ * x = -x$ has no non-trivial solutions. Where in addition, A is a densely defined, symmetric and positive operator.
I considered a solution to the equation $A^*x=-x$ and then I considered $0=<x, (A^*+I)x>$, my question is whether it is possible to do the next step $<(A+I)x, x>=<x, (A^*+I)x>$ and why I can do it.