Counterexamples to the discrete and continuous weighted Weiss conjectures

TL;DR

This paper constructs counterexamples to weighted versions of the Weiss conjecture in both discrete and continuous settings, showing that certain resolvent estimates do not imply admissibility for specific operator classes.

Contribution

It provides the first known counterexamples to weighted Weiss conjectures, highlighting limitations of resolvent conditions in operator admissibility.

Findings

Counterexamples for $oldsymbol{ ext{α} eq 0}$ in weighted Weiss conjectures

Normal operators serve as counterexamples for $oldsymbol{ ext{α} ext{ in } (-1,0)}$

Unilateral shift operators on Hardy space serve as counterexamples for $oldsymbol{ ext{α} ext{ in } (0,1)}$

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