If $T = A + iB$, where $A$ and $B$ are self-adjoint operators on a Hilbert space $H$, then this is said to be a Cartesian decomposition of T

Question

If $T = A + iB$, where $A$ and $B$ are self-adjoint operators on a Hilbert space $H$, then this is said to be a Cartesian decomposition of $T$

Compute $T^∗$ in terms of $A$ and $B$.

Answer 1

What kind of properties do you think the involution, $()^*:B(H)\rightarrow B(H)$, has?

What is $(T_1+T_2)^*$ equal to for operators $T_1, T_2$?

What is $(\lambda T)^*$ equal to for a scalar $\lambda$?

If $A$ is self-adjoint, what does $A^*$ equal to?