properties of the adjoint operator
Let $A$ and $B$ be linear operators^{} in a Hilbert space^{}, and let $\lambda \in \u2102$. 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 the restriction (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 ${\mathrm{(}{A}^{\mathrm{*}}\mathrm{)}}^{\mathrm{*}}\mathrm{=}\overline{A}$.
Title  properties of the adjoint operator 

Canonical name  PropertiesOfTheAdjointOperator 
Date of creation  20130322 13:48:14 
Last modified on  20130322 13:48:14 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  12 
Author  Koro (127) 
Entry type  Theorem 
Classification  msc 47A05 