1. ## Adjoint of Bounded Operators

If $\displaystyle S,T:H\rightarrow H$ are bounded operators, show that $\displaystyle (ST)^*=T^*S^*$.

I'm assuming that $\displaystyle H$ is a Hilbert space, although it doesn't say this in the question. I'm really not sure where to start with this. All I have is that since $\displaystyle S,T$ are bounded there adjoints $\displaystyle S^*$ and $\displaystyle T^*$ exist and that:

$\displaystyle <Tx,y>=<x,T^*y>$ for all $\displaystyle x\in H$,$\displaystyle y\in H$

2. Start by applying the adjoint principal to Tx, i.e.
$\displaystyle \langle S(Tx),y \rangle=\langle Tx,S^*y \rangle$

I hope you can now see the next step.

3. This kind of identities are usually proved as follows:

For any x,y in the hilbert space
$\displaystyle \langle x,(ST)^* y \rangle=\langle STx,y\rangle =\langle Tx,S^*y\rangle=\langle x,T^*S^*y\rangle$

From where

$\displaystyle \langle x,(ST)^* y \rangle -\langle x,T^*S^*y\rangle = 0$ for all $\displaystyle x,y \in \mathcal H$

Therefore $\displaystyle (ST)^*=T^*S^*$

Here it is used that if $\displaystyle \langle x,(B-C)y \rangle = 0$ for all $\displaystyle x,y \in \mathcal H$ then B=C