In the above screnshot, $X$ is a MASA in $L(H)$ and $Z\in L(H)$. I tried to prove (3.1) and met with a question.
When we take $Z =\begin{pmatrix}A & B\\C & D\end{pmatrix} $, where $A\in L(H_d), B\in L(H_c,H_d), C\in L(H_d,H_c), D\in L(H_c)$. Then we have $E_{X}(Z)=\begin{pmatrix}E_{D}(A) & 0\\0 & E_C(D)\end{pmatrix}$.
My question: what is the relationship between $E_{X}(Z)$ and $E_{D}(Z_{H_d})$? Or does there exist relationships between $E_D(A)$ and $E_{D}(Z_{H_d})$?
Since $H_d$ is not an invariant subspace for $Z$, so the $Z_{H_d}\in L(H_d,H)$, and I cannot express the $Z_{H_d}$ in terms of the 2 by 2 matrix with operator entries.