Decomposition of selfadjoint projections in Krein spaces

Abstract

Given a Hilbert space (H, ⟨ , ⟩) and a bounded selfadjoint operator B consider 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 sesquilinear 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 characterize 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.