$\beta$-admissibility of observation and control operators for hypercontractive semigroups

TL;DR

This paper proves a conjecture on the admissibility of control and observation operators for hypercontractive semigroups by linking them to weighted Bergman spaces and Hankel operators, with special focus on the unweighted case.

Contribution

It establishes the Weiss conjecture for $eta$-admissibility of operators in hypercontractive semigroups using a novel functional model approach.

Findings

Proves the Weiss conjecture for $eta$-admissibility.

Provides a representation of hypercontractive semigroups via weighted Bergman spaces.

Develops a reproducing kernel thesis for Hankel operators in this context.

Abstract

We prove a Weiss conjecture on $\beta$-admissibility of control and observation operators for discrete and continuous $\gamma$-hypercontractive semigroups of operators, by representing them in terms of shifts on weighted Bergman spaces and using a reproducing kernel thesis for Hankel operators. Particular attention is paid to the case $\gamma=2$, which corresponds to the unweighted Bergman shift.