Bistrict Plus-operators in Krein Spaces and Dichotomous Behavior of Irreversible Dynamical Systems
Victor Khatskevich, Leonid Zelenko · Birkhäuser Basel eBooks · 2000
We consider the ball of angular operators determined in a Krein space, i.e. in a Hilbert space with indefinite metric, for which both positive and negative components can be infinite-dimensional in general. The weak compactness of the image and the preimage of this ball by a fractional-linear transformation is established. This transformation is generated by a continuous linear operator, which is not continuously invertible in general. We assume that this operator is a bistrict plus-operator acting from a Krein space to an another one. We apply the above result to the study of dichotomous behavior of solutions to a non-autonomous linear differential equation with an unbounded operator coefficient in a Hilbert space. The evolution operator of this equation is not continuously invertible and the corresponding unstable subspace is of infinite dimension in general. As an example we study a diffusion process on the real axis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.