A Kre˘ın space coordinate free version of the de Branges complementary space

Let Z be a maximal nonnegative subspace of a Kre˘ın space X, and let X/Z be the quotient of X modulo Z. Define H(Z) = h ∈ X/Z sup −[x,x]X x ∈ h <∞ . It is proved that sup{−[x,x]X | x ∈ h} 0 for h ∈H(Z), and that H(Z) is a Hilbert space with norm h H(Z) = sup −[x,x]X x ∈ h 1/2 , which is continuously contained in X/Z, and the properties of this space are studied. Given any fundamental decomposition X =−Y [ ] U of X, the subspace Z can be written as the graph of a contraction A : U →Y. There is a natural isomorphism between X/Z and Y, and under this isomorphism the space H(Z) is mapped isometrically onto the complementary space H(A) of the range space of A studied by de Branges and Rovnyak. The space 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 Z of the Kre˘ın space X = k2+ (W) of all 2-sequences on Z+ with values in the Kre˘ınsignal space W. 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) is related to H(A). © 2008 Elsevier Inc. All rights reserved