α-Admissibility of Observation Operators in Discrete and Continuous Time

Abstract.

In this paper a weighted form of the Weiss conjecture is studied. For certain weights, the conjecture is shown to hold for normal contraction operators related to discrete time linear systems. This is proved by an application of the Carleson measure theorem for weighted Dirichlet spaces. The result for discrete time systems is used to show that a weighted form of the Weiss conjecture holds for normal operators generating bounded C₀-semigroups. Previously, weighted admissibility has been characterised for generators of analytic semigroups. No such assumption of analyticity is made here. Additionally, results are presented regarding weighted Carleson measures, fractional powers of normal operators and weighted composition operators.