Characterization of Orthogonal Projectors

Let H be a Hilbert space and M be a closed linear subspace of H. Then by projection theorem H = M ⊕ M ⊥ . This theorem suggests that the result has something to do about a notion in Hilbert spaces which is analogous to and a generalization of the familiar idea of Orthogonal or perpendicular projection of a vector in R2 or R3 upon a linear subspace of R2 or R3 respectively. In this paper we give a complete operator characterization of orthogonal projections.Specifically we show that P is an orthogonal projector onto RP = M if and only if P is self-adjoint and idempotent. We also consider the algebraic formulation of invariance, reduction, orthocomplementation and orthogonality.