Decomposition of Selfadjoint Projections in Krein Spaces

Given a Hilbert space (H, 〈 , 〉) and a bounded selfadjoint operator B con-sider the sesquilinear form over H induced by B, 〈x, y 〉B = 〈Bx, y 〉 , x, y ∈ H. A bounded operator T is B-selfadjoint if it is selfadjoint respect to this sesquilin-ear form. We study the set P(B,S) of B-selfadjoint projections with range S, where S is a closed subspace of H. We state several conditions which character-ize the existence of B-selfadjoint projections with a given range; among them certain decompositions of H, R(|B|) and R(|B|1/2). We also show that every B-selfadjoint projection can be factorized as the product of a B-contractive, a B-expansive and a B-isometric projection. Finally two different formulas for B-selfadjoint projections are given