The following screenshot is from J. C. Bourin and E. Y. Lee's paper "Pinchings and positive linear maps", J. Funct. Anal. 270, No. 1, 359-374 (2016), [MR3419765](https://mathscinet.ams.org/mathscinet-getitem?mr=MR3419765), [Zbl 1345.46050](https://zbmath.org/?q=an%3A1345.46050). When reading the proof of Corollary 3.6, I met with some problems.

**Notation**: $W_e()$ denotes the essential numerical range of an operator in $L(H)$.

How to show that 
$$
W_e\left(\begin{pmatrix}I & 0\\R_{\epsilon}& 0\end{pmatrix}\right)=W_e\left(\bigoplus\limits_{n=1}^{\infty}\begin{pmatrix}1 & 0\\a_n& 0\end{pmatrix}\right)\;?
$$ 
My thought: if we find a unitary operator $U\in L(H)$ such that 
$$
U\begin{pmatrix}I & 0\\R_{\epsilon}& 0\end{pmatrix}U^*=\bigoplus\limits_{n=1}^{\infty}\begin{pmatrix}1 & 0\\a_n& 0\end{pmatrix}\;,
$$ 
then the above conclusion holds, but how to construct the needed unitary operator?


 [![enter image description here][1]][1]


  [1]: https://i.sstatic.net/BLeCr.jpg