Counterexamples to the discrete and continuous weighted Weiss conjectures

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 $H^2(\mathbb{D})$.