If T is a linear operator on an inner product space V and $\displaystyle U_{1}$=T+T* and $\displaystyle U_{2}$=TT*, how can I show that $\displaystyle U_{1}=U_{1}^*$ and $\displaystyle U_{2}=U_{2}^*$?
If T is a linear operator on an inner product space V and $\displaystyle U_{1}$=T+T* and $\displaystyle U_{2}$=TT*, how can I show that $\displaystyle U_{1}=U_{1}^*$ and $\displaystyle U_{2}=U_{2}^*$?
since $\displaystyle (T^*)^*=T,$ we have: $\displaystyle <U_2x,y>=<TT^*x,y>=<T^*x,T^*y>=<x,TT^*y>=<x,U_2y >.$ the proof for $\displaystyle U_1$ is the same.