I have this problem I am working on:
$H$ is an inner product space with inner product $(\cdot , \cdot)$ over a subfield of the complex numbers.
Suppose $T\in L(H,H)$ has an adjoint $T^*$. Let R=$T+T^*$ and S=$T-T^*$. I'm supposed to show that R and S are normal operators and that $T ∘T^*$ and $T^*∘T$ are self adjoint.
So for the first part, to show they are normal operators, this is what I have:
For $R$: Need to show $(R(\alpha),\beta)=(\alpha,R(\beta))$ $$\implies ((T+T^*)(\alpha),\beta)=(T(\alpha)+(T^*(\alpha),(\beta))$$
I did something similar for $S$. Did I show that they are normal operators or that they are self adjoint? How do I show that $T \circ T^*$ and $T^*\circ T$ are self adjoint? Thanks!