The pseudo-regularity of the range of orthogonal projections in Krein spaces

Let P, Q be two orthogonal projections and J be a symmetry such that \(JP=QJ\). Based on the block operator technique and Halmos’ CS decomposition, we devote to characterizing the pseudo-regularity of \({\mathcal {R}}(P)\) and \({\mathcal {R}}(Q)\). It is given the J-projection onto a regular complement of \({\mathcal {R}}(P)^{\circ }\) in \({\mathcal {R}}(P)\) (resp. \({\mathcal {R}}(Q)^{\circ }\) in \({\mathcal {R}}(Q)\)). Furthermore, the sets of J-normal projections onto \({\mathcal {R}}(P)\) and \({\mathcal {R}}(Q)\) are obtained.