Let H_{1} be a closed subspace of the Hilbert space H and H_{2} its orthogonal complement in H. Then any vector ψ ∈ H can be decomposed into its components belonging to H_{1}, respectively H_{2} as follows:
ψ = ψ_{1} + ψ_{2}, ⟨ψ_{1}, ψ_{2}⟩ = 0.
'''Definition: The linear operator Q_{1} on H_{1} such that '''
- Q_{1}ψ = ψ_{1}, D(Q_{1}) = H,
'''is said to be a projector on H_{1}. '''
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 H_{2}.
The operators Q_{1} and Q_{2} have the following properties:
- They are self-adjoint;
- Their eigenspaces are H_{1} and 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_{1}Q_{2} = Q_{2}Q_{1} = 0.
We then have the following
Definition: The linear, bounded, self-adjoint, idempotent operators Q_{1} and Q_{2} on H are projectors on the orthogonal subspaces H_{1} and H_{2} 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_{jk}, j, k ∈ I,
∑_{j ∈ I}Q_{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 ∈ 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_{Δ}^{2} = Q_{Δ}.
The closed subspace of H on which Q_{Δ} projects is then H_{Δ} = ⊕ _{j ∈ Δ}H_{j}.
Category:Linear Algebra category:Handbook of Quantum Information