Projection Operator

an operator in n-dimensional Euclidean or infinite-dimensional Hilbert space that associates every vector x with its projection on some fixed subspace. For example, if H is a space of square integrable functions f(t) on the closed interval [a, b] and x(t) is the characteristic function of some closed interval [c, d] within [a, b], the mapping f(t) → X(t)f(t) is a projection operator that projects all of H onto the subspace of functions that vanish outside [c, d]. Every projection operator P is self-adjoint and satisfies the condition P² = P. Conversely, if P is a self-adjoint operator and P² = P, P is a projection operator. The concept of a projection operator plays an important role in the spectral analysis of linear operators in Hilbert space.