A Krein Space Interpretation of the Kalman-Yakubovich-Popov Lemma

In this note we give a Krein space interpretation of the celebrated Kalman-Yakubovich-Popov (KYP) Lemma by introducing state-space models driven by inputs that lie in an indefinite-metric space. Such state-space models can be considered as generalizations of standard stochastic statespace models driven by stationary stochastic processes (that lie in a definite, or so-called Hilbert, space). In this framework, the KYP lemma corresponds to a certain decomposition in Krein space.