For semigroups and for bounded operators we introduce the new notion of Bergman distance. Systems with a finite Bergman distance share the same stability properties, and the Bergman distance is preserved under the Cayley transform. This way, we get stability results in continuous and discrete time. As an example, we show that bounded perturbations lead to pairs of semigroups with finite Bergman distance. This is extended to a class of Desch–Schappacher perturbations.
- Discrete time
- Cayley transform
- Continuous time
Besseling, N. C., & Zwart, H. J. (2010). Stability analysis in continuous and discrete time, using the Cayley transform. Integral equations and operator theory, 68(4), 487-502. https://doi.org/10.1007/s00020-010-1805-8