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?


1 Answer 1


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 $(A^*+I)y=0$.


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