Let be $V$ a vector space over the complex numbers with inner product. Let be $T$ and $S$ linear operators in $V$ with adjoint operators $T^*$, and $S^*$ respectively.
I know that it satisfies this:
$\left< T(x),y \right>=\left< x,T^{*}(y) \right>$
$\left< S(x),y \right>=\left< x,S^{*}(y) \right>$ $\forall x,y \in V$
But, How can I prove that the adjoint operator of $ST$ is $(ST)^{*}$?
And how can I prove that $(ST)^{*}=T^{*}S^{*}$?
Let's take the easy one first:
There is really nothing to prove here. "The adjoint operator of $S$" is shortened to $S^*$, and "The adjoint operator of $TS$" is shortened to $(TS)^*$. That's all there is to it.
By using the definition of adjoint, of course. For any $x,y\in V$ we have $$ \langle x, (ST)^*y\rangle=\langle STx,y\rangle\\=\langle S(Tx), y\rangle = \langle Tx, S^*y\rangle\\=\langle x, T^*S^*y\rangle $$ Each and every step is justified from the definition: One can move an operator from acting on the left vector to acting on the right vector by taking the adjoint (and vice versa). And since we now have shown that $\langle x, (ST)^*y\rangle=\langle x, T^*S^*y\rangle$ for all $x,y\in V$, the operators themselves must be equal.