Let $H$ be an infinite dimensional separable complex Hilbert space with orthonormal basis.
Let $W \in B(H)$ be a unilateral forward weighted shift. i.e. $We_n=w_n e_{n+1}$, where $(w_n) \in \ell^{\infty}$.
Show that there exists a unitary operator $U\in B(H)$ so that $U^*WU$ is a unilateral weighted shift with weight sequence $(|w_n|)_n$.
My work:
I have $ Ue_n= \begin{cases} \frac{w_n}{|w_n|}e_n,& \text{if } w_n \ne 0\\ e_n, & \text{if } w_n=0 \end{cases} $
I showed that $U$ is unitary.
Now I want to show that $U^*WU$ is a unilateral weighted shift with weight sequence $(|w_n|)_n$.
So in the case $w_n \ne 0$, I have
$U^*WU e_n= U^*W(\frac{w_n}{|w_n|}e_n) = \frac{w_n}{|w_n|}U^*W(e_n)= \frac{w_n}{|w_n|}U^*w_n(e_{n+1}) = \frac{w_n}{|w_n|}w_nU^*(e_{n+1}) = \frac{w_n}{|w_n|}w_n\frac{w_{n+1}}{|w_{n+1}|}e_{n+1} = |w_n|e_{n+1}$.
I'm not sure if the last equality is correct, because I just sort of forced it in the last step...
If not, could someone let me know how to fix it?
Also, any other suggestions will be highly appreciated too!
Thank you.
Let's define a sequence $u$ such that: $$ u_0 = 0\\ u_{n+1}=u_n+\arg w_n $$ And for the operator $U$ holds: $Ue_n = e^{iu_n}e_n$
Now let's proove that it works:$$ U^*WUe_n = e^{iu_n}U^*We_n=e^{iu_n}w_nU^*e_{n+1}=e^{iu_n}w_ne^{-iu_{n+1}}e_{n+1}= \\=\exp(iu_n+ln|w_n|+i\arg w_n-iu_{n+1})e_{n+1}=|w_n|\exp(iun+i\arg w_n - iu_n-i\arg w_n)e_{n+1}=|w_n|e_{n+1} $$