Unitary operator statement

Question

I encountered this statement in the book $$W(U^{*}TU)=W(T)$$ for any unitary U where $W(T)$ is the numerical range. Definition of the numerical range is $$W(T)=\{\langle Tx,x\rangle |x\in H,\|x\|=1\}$$ I know that $U$ is unitary which means $$UU^{*}=I=U^{*}U$$ Trying to prove this should go like this $\langle U^{*}TUx,x\rangle =\langle TUx,Ux\rangle =\langle Ux,T^{*}Ux\rangle =\langle x,U^{*}T^{*}Ux\rangle $ and I dont know how to proceed here. It should simplify to just $\langle Tx,x\rangle $...I think...

Any help would be great, thanks!

Answer 1

What you have is that $\|Ux\|=1$ if and only if $\|x\|=1$. So the unitary is an automorphism of the set $\{x:\ \|x\|=1\}$. Hence \begin{align} W(T)&=\{\langle Tx,x\rangle:\ \|x\|=1\}\\ &=\{\langle TUx,Ux\rangle:\ \|Ux\|=1\}\\ &=\{\langle TUx,Ux\rangle:\ \|x\|=1\}\\ &=\{\langle U^*TUx,x\rangle:\ \|x\|=1\}\\ &=W(U^*TU). \end{align}