properties of the adjoint operator

Let $A$ and $B$ be linear operators in a Hilbert space,and let $\lambda \in ℂ$. Assuming all the operators involved are densely defined, the following properties hold:

1. If $A^{-1}$ exists and is densely defined, then ${(A^{-1})}^{*}={(A^{*})}^{-1}$;

2. ${(\lambda A)}^{*}=\overline{\lambda }{A}^{*}$;

3. $A\subset B$ implies $B^{*}\subset A^{*}$;

4. $A^{*}+B^{*}\subset {(A+B)}^{*}$;

5. $B^{*}{A}^{*}\subset {(AB)}^{*}$;

6. ${(A+\lambda I)}^{*}=A^{*}+\overline{\lambda }I$;

7. $A^{*}$ is a closed operator.

Remark. The notation $A\subset B$ for operators means that $B$ is an of $A$, i.e. $A$ is therestriction (http://planetmath.org/RestrictionOfAFunction) of $B$ to a smaller domain.

Also, we have the following

Proposition 1

If $A$ admits a closure (http://planetmath.org/ClosedOperator) $\overline{A}$, then $A^{\mathrm{*}}$ is densely defined and ${\mathrm{(}{A}^{\mathrm{*}}\mathrm{)}}^{\mathrm{*}}\mathrm{=}\overline{A}$.