Counterexamples to the Discrete and Continuous Weighted Weiss Conjectures

Abstract

Counterexamples are presented to weighted forms of the Weiss conjecture in discrete and continuous time. In particular, for certain ranges of $\alpha$, operators are constructed that satisfy a given resolvent estimate, but fail to be $\alpha$-admissible. For $\alpha\in(-1,0)$ the operators constructed are normal, while for $\alpha\in(0,1)$ the operator is the unilateral shift on the Hardy space $H^2(\mathbb{D})$.