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 extension of $A$ i.e. $A$ is the restriction of $B$ to a smaller domain.

Also, we have the following