# properties of the adjoint operator

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

1. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

6. 6.

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

7. 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 $A^{*}$ is densely defined and $(A^{*})^{*}=\overline{A}$.

Title properties of the adjoint operator PropertiesOfTheAdjointOperator 2013-03-22 13:48:14 2013-03-22 13:48:14 Koro (127) Koro (127) 12 Koro (127) Theorem msc 47A05