Example of a rank-one operator to satisfy conditions about the spectrum $U$ is defined to be a bilateral shift operator such that $Ue_n= u_ne_{n+1}$ for $n\in \mathbb{Z}$, where $u_n \ne 1$.
I am looking for an example of $U$ and a rank-one operator $T \in \mathcal{B}(\ell^2(\mathbb{Z}))$ so that $\sigma (U) = \{ z \in \mathbb{C} : |z| = 1 \}$ and $\sigma(U+T)=  \{ z \in \mathbb{C} : |z| \le 1 \}$.
Some thoughts:
I showed that for any bilateral shift $U$ with $Ue_n=u_ne_{n+1}$ for $n \in \mathbb{Z}$ such that $|u_n|=1$, we have $ \sigma (U) = \{ z \in \mathbb{C} : |z| = 1 \}$.
Also, I saw it from here The spectrum of the operators $\sigma(U+T) = \sigma(U) \cup \sigma(T)  $.
So I wonder if I could find some rank one operator to satisfy the condition based on that.
But other suggestions will also be great.
Thank you.
 A: Let $V$ be the standard bilateral shift on $\ell ^2(\mathbb Z)$,  namely the unitary operator $V$ such that $Ve_n=e_{n+1}$,  and let $T$ be the rank-one operator
given by
$$
  T(x)= -\langle x,e_0\rangle e_1, \quad\forall x\in  \ell ^2(\mathbb Z).
  $$
Then   $R:= V+T$ becomes the operator given by
$$
  R(e_n) = \left\{\matrix{
  e_{n+1}, & \text { if } n\neq 1, \cr
  0 , & \text { if } n=0.
  }\right.
  $$
One then sees that $R$ leaves invariant the decomposition
$$
  \ell ^2(\mathbb Z)=\ell ^2(\mathbb Z_-)\oplus \ell ^2(\mathbb Z_+^*),
  $$
where $\mathbb Z_-$ stands for the non-positive integers and $\mathbb Z_+^*$,  for the strictly positive ones.
The restriction of $R$ to $\ell ^2(\mathbb Z_+^*)$ is then  the so called unilateral shift $S$, while the restriction of $R$
to $\ell ^2(\mathbb Z_-)$
is seen to be unitarily equivalent to $S^*$.  We then conclude that $R$ is unitarily equivalent to $S\oplus S^*$, whose spectrum
is given by
$$
  \sigma (R) = \sigma (S) \cup  \sigma (S^*) = \sigma (S) \cup  \overline{ \sigma (S)}.
  $$
Since $\sigma (S)$ is the unit disk,  we see that   $\sigma (R)$ is the unit disk as well.
Now,  the OP has asked for a version of the bilateral shift where $Ue_n=u_ne_{n+1}$,  with $u_n\neq 1$,  so we may take
$U=-V$,  i.e.  take all of the $u_n=-1$,  the requested rank-one operator being $-T$.
