Automorphisms of Grassmannians

For a complex vector space V of dimension n, the group of holomorphic automorphisms of the Grassmannian Gr(p,V) can be identified with the subgroup of PG1( ᵖV) preserving the Grassmannian. Using this, Chow showed Aut (Gr(p,V)) = PGl (V) for n ≠ 2p, and PG1(V) is a normal subgroup of index 2 in Aut(Gr(p,V)) for $n = 2p$. We prove a version of Chow’s result for a separable Hilbert space H. Theorem. PGl (H) is the subgroup of PGl ( ᵖH) which preserves Gr(p,H). That is, if R is an invertible linear operator on ᵖH which preserves decomposable p-vectors, then there exists S, an invertible linear operator on H, such that R = ᵖS.