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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?

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    $\begingroup$ What is $R_\epsilon$? Also, you are displaying an image taken from a paper. What is the paper? $\endgroup$
    – LSpice
    Commented May 23, 2022 at 16:10
  • $\begingroup$ $R_\epsilon$ is a positive diagonalizable operator . $\endgroup$
    – mathbeginner
    Commented May 23, 2022 at 22:22
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    $\begingroup$ I’m voting to close this question until more information is given about the book/paper that is being followed $\endgroup$
    – Yemon Choi
    Commented May 24, 2022 at 20:43

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