# Projector

Let $\; \mathcal\left\{H\right\}_1$ be a closed subspace of the Hilbert space $\; \mathcal\left\{H\right\}$ and $\; \mathcal\left\{H\right\}_2$ its orthogonal complement in $\; \mathcal\left\{H\right\}$. Then any vector $\; \psi \in \mathcal\left\{H\right\}$ can be decomposed into its components belonging to $\; \mathcal\left\{H\right\}_1$, respectively $\; \mathcal\left\{H\right\}_2$ as follows: :$\; \psi = \psi_1 + \psi_2, \quad \langle \psi_1,\psi_2\rangle =0 .$ '''Definition: The linear operator $\; Q_1$ on $\; \mathcal\left\{H\right\}_1$ such that ''' : $\; Q_1\psi = \psi_1 , \quad \mathcal\left\{D\right\}\left(Q_1\right)= \mathcal\left\{H\right\} ,$ '''is said to be a ''projector'' on $\; \mathcal\left\{H\right\}_1$. ''' Having defined the operator $\; Q_2 = \mathbf\left\{1\right\}- Q_1$ such that : $\; Q_2\psi =\psi - \psi_1 = \psi_2 , \quad \mathcal\left\{D\right\}\left(Q_2\right)= \mathcal\left\{H\right\} ,$ the latter is a projector on $\; \mathcal\left\{H\right\}_2$. The operators $\; Q_1$ and $\; Q_2$ have the following properties: # They are self-adjoint; # Their eigenspaces are $\; \mathcal\left\{H\right\}_1$ and $\; \mathcal\left\{H\right\}_2$ respectively corresponding to the eigenvalues $\; \left(1,0\right)$ respectively $\; \left(0,1\right)$; # They have a complete set of eigenvectors and their descrete spectrum is $\; Sp\left(Q_a\right) =\\left\{1,0\\right\}, \; a=1,2$; # 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 $\; \mathcal\left\{H\right\}$ are projectors on the orthogonal subspaces $\; \mathcal\left\{H\right\}_1$ and $\; \mathcal\left\{H\right\}_2$ respectively.''' These results can be generalized to the case of more than two projectors, as follows. '''Definition: Let $\; \\left\{Q_j : j \in I\\right\}$, with $\; I$ an arbitrary finite or countably infinite subset, be a set of projectors such that :$\; Q_jQ_k = \delta_\left\{jk\right\} , \quad j,k \in I ,$ :$\; \sum_\left\{j \in I\right\}Q_j = \mathbf\left\{1\right\} ,$''' '''then we will say that $\; \\left\{Q_j : j \in I\\right\}$ is a complete set of orthogonal projectors.''' Each projector $\; Q_j$ belongs to a closed subspace $\; \mathcal\left\{H\right\}_j$ of the Hilbert space $\; \mathcal\left\{H\right\}$, and these subspaces give a complete orthogonal decomposition of $\; \mathcal\left\{H\right\}$: :$\; \mathcal\left\{H\right\} = \oplus_\left\{j \in I\right\} \mathcal\left\{H\right\}_j , \quad \mathcal\left\{H\right\}_j = Q_j\mathcal\left\{H\right\} .$ From completeness it follows that $\; \psi= \sum_\left\{j \in I\right\}Q_j\psi$, with $\; Q_j\psi \in \mathcal\left\{H\right\}_j$, and therefore each component of $\; \psi$ in the subspace $\; \mathcal\left\{H\right\}_j$ is given by the projection of $\; \psi$ in $\; \mathcal\left\{H\right\}_j$: $\; \psi_j=Q_j\psi$. Moreover, if $\; \Delta$ is a subset of $\; I$, then the operator $\; Q_\left\{\Delta\right\}= \sum_\left\{j \in \Delta\right\}Q_j$ is a projector. In fact it can be easily verified that the following properties are satisfied: # $\; \mathcal\left\{D\right\}\left(Q_\Delta\right)=\mathcal\left\{H\right\}$; # $\; Q_\Delta^\dagger=Q_\Delta$; # $\; Q_\Delta^2=Q_\Delta$. The closed subspace of $\; \mathcal\left\{H\right\}$ on which $\; Q_\left\{\Delta\right\}$ projects is then $\; \mathcal\left\{H\right\}_\left\{\Delta\right\}= \oplus_\left\{j \in \Delta\right\}\mathcal\left\{H\right\}_j$. Category:Linear Algebra category:Handbook of Quantum Information