Self-adjointness Criteria and Essential Self-adjointness

Prove that each of the following conditions is equivalent to the essential selfadjointness of an operator $A$:

1. $A^*$ is selfadjoint;

2. $\bar A=A^*$.

In which of the following cases is the operator $A=i\, d/dx$ in $H=L^{2}\left({0,1}\right)$ symmetric, essentially selfadjoint, or selfadjoint:

Suppose that $H=l_2(\mathbb {C})$, $D_A$ consists of all the finitely nonzero sequences with zero sum, and $A$ is the operator given by the matrix $A=(a_{jk})$, where $a_{jk}=i\operatorname {sign}\left(j-k\right)$.

Let $A_i$, $i=1,2$, be the operator of multiplication by $x$ in $L^{2}\left({\mathbb {R},dx}\right)$ with domains $D_{A_i}$. Suppose that $A_1$ and $A_2$ are essentially selfadjoint. Is it possible that the subspaces $D_{A_1}$ and $D_{A_2}$ have zero intersection?

Suppose that the operator $A$ is densely defined and positive (see Problem 643). Prove that $A$ is essentially selfadjoint if and only if $\ker (A^*+1)=\left\{ 0\right\}$.

Suppose that $H_1$ and $H_2$ are Hilbert spaces, and $H_1\otimes H_2$ their Hilbert tensor product. Prove that if $A_1$ and $A_2$ are selfadjoint operators in $H_1$ and $H_2$, respectively, then the operator $A_1\otimes 1+1\otimes A_2$, with domain $D_{A_1}\otimes D_{A_2}$, is essentially selfadjoint in $H$.