Let  ℋ₁ be a closed subspace of the Hilbert space  ℋ and  ℋ₂ its orthogonal complement in  ℋ. Then any vector  ψ ∈ ℋ can be decomposed into its components belonging to  ℋ₁, respectively  ℋ₂ as follows:

 ψ = ψ₁ + ψ₂,  ⟨ψ₁, ψ₂⟩ = 0.

Definition: The linear operator  Q₁ on  ℋ₁ such that

 Q₁ψ = ψ₁,  𝒟(Q₁) = ℋ,

is said to be a projector on  ℋ₁.

Having defined the operator  Q₂ = 1 − Q₁ such that

 Q₂ψ = ψ − ψ₁ = ψ₂,  𝒟(Q₂) = ℋ,

the latter is a projector on  ℋ₂.

The operators  Q₁ and  Q₂ have the following properties:

1. They are self-adjoint;
2. Their eigenspaces are  ℋ₁ and  ℋ₂ respectively corresponding to the eigenvalues  (1, 0) respectively  (0, 1);
3. They have a complete set of eigenvectors and their descrete spectrum is  Sp(Q_a) = {1, 0}, a = 1, 2;
4. They are idempotent, i.e. such that  Q_a² = Q_a;
5. They are orthogonal to each other, i.e. such that  Q₁Q₂ = Q₂Q₁ = 0.

We then have the following

Definition: The linear, bounded, self-adjoint, idempotent operators  Q₁ and  Q₂ on  ℋ are projectors on the orthogonal subspaces  ℋ₁ and  ℋ₂ respectively.

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

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

 Q_jQ_k = δ_jk,  j, k ∈ I,

 ∑_j ∈ IQ_j = 1,then we will say that  {Q_j : j ∈ I} is a complete set of orthogonal projectors.'''

Each projector  Q_j belongs to a closed subspace  ℋ_j of the Hilbert space  ℋ, and these subspaces give a complete orthogonal decomposition of  ℋ:

 ℋ = ⊕_j ∈ Iℋ_j,  ℋ_j = Q_jℋ.

From completeness it follows that  ψ = ∑_j ∈ IQ_jψ, with  Q_jψ ∈ ℋ_j, and therefore each component of  ψ in the subspace  ℋ_j is given by the projection of  ψ in  ℋ_j:  ψ_j = Q_jψ.

Moreover, if  Δ is a subset of  I, then the operator  Q_Δ = ∑_j ∈ ΔQ_j is a projector. In fact it can be easily verified that the following properties are satisfied:

1.  𝒟(Q_Δ) = ℋ;
2.  Q_Δ^† = Q_Δ;
3.  Q_Δ² = Q_Δ.

The closed subspace of  ℋ on which  Q_Δ projects is then  ℋ_Δ = ⊕_j ∈ Δℋ_j.

Category:Linear Algebra category:Handbook of Quantum Information