Notation: $W_e()$ denotes the essential numerical range of an operator in $L(H)$ and $\Bbb D$ is the unit disk of $\Bbb C$.

I tried to prove the conclusion : $\Bbb D\subset W_e(Q)$ iff $\Bbb D\subset W_e(Q_{H_{ns}})$.

It is easy to check that the "if part".

But for the converse direction, I met with some problems. For any $\lambda\in \Bbb D$ ,there exists a basis $\{e_n\}_{n=1}^{\infty}\cup\{f_n\}_{n=1}^{\infty}$ such that

$$\lambda=\lim_{n}\langle\begin{pmatrix} e_n \\ f_n \end{pmatrix}, (P\oplus \begin{pmatrix}I & 0\\R& 0\end{pmatrix}) \begin{pmatrix} e_n \\ f_n \end{pmatrix}\rangle$$, how to show that for any $\lambda\in \Bbb D$, there is a basis $\{g_n\}_{n=1}^{\infty}$such that

$\lambda=\lim_{n}\langle g_n, \begin{pmatrix}I & 0\\R& 0\end{pmatrix} g_n\rangle$,