# Projector

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* = *ψ*1 + *ψ*2, ⟨*ψ*1, *ψ*2⟩ = 0.

'''Definition: The linear operator *Q*1 on H1 such that '''

*Q*1

*ψ*=

*ψ*1, D(

*Q*1) = H,

'''is said to be a *projector* on H1. '''

Having defined the operator *Q*2 = **1** − *Q*1 such that

*Q*2

*ψ*=

*ψ*−

*ψ*1 =

*ψ*2, D(

*Q*2) = H,

the latter is a projector on H2.

The operators *Q*1 and *Q*2 have the following properties:

- They are self-adjoint;
- Their eigenspaces are H1 and H2 respectively corresponding to the eigenvalues (1, 0) respectively (0, 1);
- They have a complete set of eigenvectors and their descrete spectrum is
*S**p*(*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*1*Q*2 =*Q*2*Q*1 = 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**j**Q**k* = *δ**j**k*, *j*, *k* ∈ *I*,

∑*j* ∈ *I**Q**j* = **1**, ''' **then we will say that { Qj : 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* ∈ *I*H*j*, H*j* = *Q**j*H.

From completeness it follows that *ψ* = ∑*j* ∈ *I**Q**j**ψ*, with *Q**j**ψ* ∈ 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**j**ψ*.

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

- D(
*Q*Δ) = H; -
*Q*Δ † =*Q*Δ; -
*Q*Δ2 =*Q*Δ.

The closed subspace of H on which *Q*Δ projects is then HΔ = ⊕ *j* ∈ ΔH*j*.