A Kreĭn space coordinate free version of the de Branges complementary space

Let ZZ be a maximal nonnegative subspace of a Kreĭn space XX, and let X/ZX/Z be the quotient of XX modulo ZZ. DefineH(Z)={h∈X/Z|sup{−[x,x]X|x∈h}<∞}.H(Z)={h∈X/Z|sup{−[x,x]X|x∈h}<∞}. It is proved that sup{−[x,x]X|x∈h}⩾0sup{−[x,x]X|x∈h}⩾0 for h∈H(Z)h∈H(Z), and that H(Z)H(Z) is a Hilbert space with norm‖h‖H(Z)=(sup{−[x,x]X|x∈h})1/2,‖h‖H(Z)=(sup{−[x,x]X|x∈h})1/2, which is continuously contained in X/ZX/Z, and the properties of this space are studied. Given any fundamental decomposition X=−Y[∔]UX=−Y[∔]U of XX, the subspace ZZ can be written as the graph of a contraction A:U→YA:U→Y. There is a natural isomorphism between X/ZX/Z and YY, and under this isomorphism the space H(Z)H(Z) is mapped isometrically onto the complementary space H(A)H(A) of the range space of A   studied by de Branges and Rovnyak. The space H(Z)H(Z) is used as state space in a construction of a canonical passive state/signal shift realization of a linear observable and backward conservative discrete time invariant state/signal system with a given passive future behavior, equal to a given maximal nonnegative right-shift invariant subspace ZZ of the Kreĭn space X=k+2(W) of all ℓ2ℓ2-sequences on Z+Z+ with values in the Kreĭn signal space WW. This state/signal realization is related to the de Branges–Rovnyak model of a linear observable and backward conservative scattering input/state/output system whose scattering matrix is a given Schur class function in the same way as H(Z)H(Z) is related to H(A)H(A).