Minimal Passive Realizations of Generalized Schur Functions in Pontryagin Spaces

Abstract

Passive discrete-time systems in Pontryagin space setting are investigated. In this case the transfer functions of passive systems, or characteristic functions of contractive operator colligations, are generalized Schur functions. The existence of optimal and * -optimal minimal realizations for generalized Schur functions are proved. By using those realizations, a new definition, which covers the case of generalized Schur functions, is given for defects functions. A criterion due to D.Z. Arov and M.A. Nudelman, when all minimal passive realizations of the same Schur function are unitarily similar, is generalized to the class of generalized Schur functions. The approach used here is new; it relies completely on the theory of passive systems.

Generalized Schur–Nevanlinna functions and their realizations

“…A special case, where the system is passive, i.e. the system operator is contractive, is proved in [32,Lemma 2.4]; see also the proof of [34,Theorem 2.2]. The proofs given therein can be applied word by word to get the next result, since the existence of defect operator is guaranteed by Theorem 3.2.…”

“…In the case where all the spaces are Hilbert spaces, it is well known; see for instance [8,Proposition 8], that the transfer function of the passive system is an ordinary Schur function. In general case where X , U and Y are Pontryagin spaces such that U and Y have the same negative index, the transfer function of the passive system Σ = (T Σ ; X , U; κ) is a generalized Schur function, with the index not larger that the negative index of the state space [32,Proposition 2.4]. Conversely, every θ ∈ S κ (U, Y) has a realization of the form (1.5), and the realization can be chosen such that it is controllable isometric (observable co-isometric, simple conservative, minimal passive) […”

“…That is, Z is bounded injective self-adjoint operator. Moreover, an optimal ( *optimal) minimal passive realization of θ ∈ S κ (U, Y) is unique up to unitary similarity [32,Theorem 3.5]. Therefore, the mapping Z is unique up to unitary equivalence, and the properties of Z in Theorem 3.10 below do not depend of the choice of a * -optimal minimal passive realization of θ.…”

“…Theory of passive systems and generalized Schur functions in the case where U and Y are Pontryagin spaces with the same negative index, was studied by the author in [32]. On the other aspects, in the case where U and Y are finite dimensional, the class S κ (U, Y) is closely related to generalized Potapov class and generalized J -inner functions; see for instance [1,6,8,21,37].…”

“…On the other hand, every θ ∈ S κ (U, Y) has a minimal passive realization Σ, and it can be chosen such that it is optimal or * -optimal [32,Theorem 3.5]; for the case where U and Y are Hilbert spaces, see also [34,Theorem 5.3]. For a symmetric θ ∈ S κ (U), these realizations have special properties.…”