The following screenshot is from J.C.Bourin and E.Y.Lee's paper"Pinchings and positive linear maps". 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 uniatry operator?

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, Zbl 1345.46050. 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?

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 uniatry operator?

The following screenshot is from J.C.Bourin and E.Y.Lee's paper"Pinchings and positive linear maps". 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 uniatry operator?

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

How to show that $W_e(\begin{pmatrix}I & 0\\R_{\epsilon}& 0\end{pmatrix})$=$W_e(\oplus_{n=1}^{\infty}\begin{pmatrix}1 & 0\\a_n& 0\end{pmatrix})$

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

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 uniatry operator?