Definition:Matrix Notation (Bounded Linear Operator)

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Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a bounded linear operator on $H$.

Let $M$ be a closed linear subspace of $H$; denote by $M^\perp$ its orthocomplement.

Observe that $M \oplus M^\perp \cong H$ by Direct Sum of Subspace and Orthocomplement.

The associated isomorphism $U$ induces a bounded linear operator $A'$ on $M \oplus M^\perp$.

$A'$ can be written as the matrix:

$\begin{pmatrix} W & X \\ Y & Z \end{pmatrix}$

where $W \in B \left({M}\right)$, $X \in B \left({M^\perp, M}\right)$, $Y \in B \left({M, M^\perp}\right)$, $Z \in B \left({M^\perp}\right)$.

In the above matrix, linear transformations $W, X, Y, Z$ are uniquely determined by the requirement that:

$\begin{pmatrix} Wm + Xm^\perp \\ Ym + Zm^\perp\end{pmatrix} = \begin{pmatrix} W & X \\ Y & Z \end{pmatrix} \begin{pmatrix}m \\m^\perp \end{pmatrix} = U^{-1} \left({Ah}\right)$ whenever $h = U \left({m, m^\perp}\right)$.

By abuse of notation, one writes $A$ instead of $A'$ and calls it the matrix notation for $A$ (with respect to $M$).