Calculus of Orthogonal Projectors

Abstract

It is possible to express all geometric notions connected with closed linear subspaces in terms of algebraic properties of the orthoprojectors onto these linear spaces. In this paper, sufficient conditions for the calculus of a family of orthoprojectors in B(H) have been given with meaningful consideration of the sum, the product and difference of orthoprojectors to be a projector. This has been done by giving the algebraic formulations of orthogonality for the sum, product and difference. From the paper, it is observed that there is a natural one-to-one correspondence between the set of all closed linear subspaces of a Hilbert space H and the set of all orthoprojectors on H. This paper will help in the study of vector space with many diverse applications such as orthogonal polynomials, QR decomposition of projectors and Gram-Schmidt orthogonalization.