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Projector

Let $\; \mathcal{H}_1$ be a closed subspace of the Hilbert space $\; \mathcal{H}$ and $\; \mathcal{H}_2$ its orthogonal complement in $\; \mathcal{H}$ . Then any vector $\; \psi \in \mathcal{H}$ can be decomposed into its components belonging to $\; \mathcal{H}_1$ , respectively $\; \mathcal{H}_2$ as follows:

$\; \psi = \psi_1 + \psi_2, \quad \langle \psi_1,\psi_2\rangle =0 .$

Definition: The linear operator $\; Q_1$ on $\; \mathcal{H}_1$ such that

$\; Q_1\psi = \psi_1 , \quad \mathcal{D}(Q_1)= \mathcal{H} ,$

is said to be a projector on $\; \mathcal{H}_1$ .

Having defined the operator $\; Q_2 = \mathbf{1}- Q_1$ such that

$\; Q_2\psi =\psi - \psi_1 = \psi_2 , \quad \mathcal{D}(Q_2)= \mathcal{H} ,$

the latter is a projector on $\; \mathcal{H}_2$ .

The operators $\; Q_1$ and $\; Q_2$ have the following properties:

They are self-adjoint;
Their eigenspaces are $\; \mathcal{H}_1$ and $\; \mathcal{H}_2$ respectively corresponding to the eigenvalues $\; (1,0)$ respectively $\; (0,1)$ ;
They have a complete set of eigenvectors and their descrete spectrum is $\; Sp(Q_a) =\{1,0\}, \; a=1,2$ ;
They are idempotent, i.e. such that $\; Q_a^2 = Q_a$ ;
They are orthogonal to each other, i.e. such that $\; Q_1Q_2 = Q_2Q_1=0$ .

We then have the following

Definition: The linear, bounded, self-adjoint, idempotent operators $\; Q_1$ and $\; Q_2$ on $\; \mathcal{H}$ are projectors on the orthogonal subspaces $\; \mathcal{H}_1$ and $\; \mathcal{H}_2$ respectively.

These results can be generalized to the case of more than two projectors, as follows.

Definition: Let $\; \{Q_j : j \in I\}$ , with $\; I$ an arbitrary finite or countably infinite subset, be a set of projectors such that

$\; Q_jQ_k = \delta_{jk} , \quad j,k \in I ,$

$\; \sum_{j \in I}Q_j = \mathbf{1} ,$

then we will say that $\; \{Q_j : j \in I\}$ is a complete set of orthogonal projectors.

Each projector $\; Q_j$ belongs to a closed subspace $\; \mathcal{H}_j$ of the Hilbert space $\; \mathcal{H}$ , and these subspaces give a complete orthogonal decomposition of $\; \mathcal{H}$ :

$\; \mathcal{H} = \oplus_{j \in I} \mathcal{H}_j , \quad \mathcal{H}_j = Q_j\mathcal{H} .$

From completeness it follows that $\; \psi= \sum_{j \in I}Q_j\psi$ , with $\; Q_j\psi \in \mathcal{H}_j$ , and therefore each component of $\; \psi$ in the subspace $\; \mathcal{H}_j$ is given by the projection of $\; \psi$ in $\; \mathcal{H}_j$ : $\; \psi_j=Q_j\psi$ .

Moreover, if $\; \Delta$ is a subset of $\; I$ , then the operator $\; Q_{\Delta}= \sum_{j \in \Delta}Q_j$ is a projector. In fact it can be easily verified that the following properties are satisfied: