Two canonical passive state/signal shift realizations of passive discrete time behaviors

Abstract

A discrete time invariant linear state/signal system Σ with a Hilbert state space X and a Kreĭn signal space W has trajectories ( x ( ⋅ ) , w ( ⋅ ) ) that are solutions of the equation x ( n + 1 ) = F ( [ x ( n ) u ( n ) ] ) , where F is a bounded linear operator from [ X W ] into X with a closed domain whose projection onto X is all of X . This system is passive if the graph of F is a maximal nonnegative subspace of the Kreĭn space − X [ ∔ ] X [ ∔ ] W . The future behavior W fut of a passive system Σ is the set of all signal components w ( ⋅ ) of trajectories ( x ( ⋅ ) , w ( ⋅ ) ) of Σ on Z + = { 0 , 1 , 2 , … } with x ( 0 ) = 0 and w ( ⋅ ) ∈ ℓ 2 ( Z + ; W ) . This is always a maximal nonnegative shift-invariant subspace of the Kreĭn space k 2 ( Z + ; W ) , i.e., the space ℓ 2 ( Z + ; W ) endowed with the indefinite inner product inherited from W . Subspaces of k 2 ( Z + ; W ) with this property are called passive future behaviors. In this work we study passive state/signal systems and passive behaviors (future, full, and past). In particular, we define and study the input and output maps of a passive state/signal system, and the past/future map of a passive behavior. We then turn to the inverse problem, and construct two passive state/signal realizations of a given passive future behavior W + , one of which is observable and backward conservative, and the other controllable and forward conservative. Both of these are canonical in the sense that they are uniquely determined by the given data W + , in contrast earlier realizations that depend not only on W + , but also on some arbitrarily chosen fundamental decomposition of the signal space W . From our canonical realizations we are able to recover the two standard de Branges–Rovnyak input/state/output shift realizations of a given operator-valued Schur function in the unit disk.