Certain Properties of Orthogonal Projections

Let P(H) be the set of orthogonal projections on a Hilbert space H. We study the properties of the set: F(P) = {Q ∈ P(H) : (P, Q) is Fredholm pair}, where (P, Q) is a Fredholm pair if PQ|R(Q) : R(Q) −→ R(P) is a Fredholm operator in B R(Q),R(P) . We describe models and factorizations for elements in F(P),which are related to the geometry of P ∈ P(H). The study ofF(P) throws new light on the geodesic structure of Q ∈ F(P). Also, we study the subsets of restricted Grassmannian: Gres(P) = {Q ∈ P(H) : (P, Q) is restricted Grassmannian}. We show that Q ∈ Gres(P) if and only if P−Q is compact. Some properties of combinations = c1P +c2Q +c3PQ +c4QP +c5PQP +c6QPQ +c7QPQP, ci ∈ C, i = 1, . . . , 7 are obtained. The multi-relations among compact, Fredholm, and restricted Grassmannian are investigated.