Certain Properties of Orthogonal Projections

Let $$\mathcal {P(H)}$$ be the set of orthogonal projections on a Hilbert space $${\mathcal {H}}$$ . We study the properties of the set: $$\begin{aligned} {\mathcal {F}}(P)=\{Q\in \mathcal {P(H)}: (P, Q)\ {\hbox {is Fredholm pair}}\}, \end{aligned}$$ where (P, Q) is a Fredholm pair if $$PQ|_{{\mathcal {R}}(Q)}:{\mathcal {R}}(Q)\longrightarrow {\mathcal {R}}(P)$$ is a Fredholm operator in $${\mathcal {B}}\big ({\mathcal {R}}(Q), {\mathcal {R}}(P)\big )$$ . We describe models and factorizations for elements in $${\mathcal {F}}(P)$$ , which are related to the geometry of $$P\in \mathcal {P(H)}$$ . The study of $${\mathcal {F}}(P)$$ throws new light on the geodesic structure of $$Q\in {\mathcal {F}}(P).$$ Also, we study the subsets of restricted Grassmannian: $$\begin{aligned} {\mathcal {G}}_{\mathrm{res}}(P)=\{Q\in \mathcal {P(H)}: (P, Q)\ {\hbox {is restricted Grassmannian}}\}. \end{aligned}$$ We show that $$Q\in {\mathcal {G}}_{\mathrm{res}}(P)$$ if and only if $$P-Q$$ is compact. Some properties of combinations $$\Gamma =c_1P+c_2Q +c_3PQ +c_4QP+c_5PQP+c_6QPQ +c_7QPQP,\quad c_i\in {\mathbb {C}},\ i=1,\ldots , 7$$ are obtained. The multi-relations among compact, Fredholm, and restricted Grassmannian are investigated.