Admissible operators and ${\mathcal H}_{\infty}$ calculus

TL;DR

This paper investigates the properties of admissible operators related to the ${ m H}_ ext{infty}$ calculus for generators of exponentially stable semigroups on Hilbert spaces, establishing conditions for boundedness and connections to observability.

Contribution

It demonstrates the existence of infinite-time admissible output operators for functions in ${ m H}_ extinfty$, characterizes when these operators are bounded, and links bounded ${ m H}_ extinfty$-calculus to observability.

Findings

Admissible operators exist for all ${ m H}_ extinfty$ functions.

Boundedness of $g(A)$ is characterized by rationality and observability.

Connections between admissibility, bounded calculus, and well-known generator classes.

Abstract

Given a Hilbert space and the generator $A$ of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any $g(-s) \in {\mathcal H}_{\infty}$ we show that there exists an infinite-time admissible output operator $g(A)$. If $g$ is rational, then this operator is bounded, and equals the "normal" definition of $g(A)$. In particular, when $g(s)=1/(s + \alpha)$, $ \alpha \in {\mathbb C}_0^+$, then this admissible output operator equals $(\alpha I - A)^{-1}$. Although in general $g(A)$ may be unbounded, we always have that $g(A)$ multiplied by the semigroup is a bounded operator for every (strictly) positive time instant. Furthermore, when there exists an admissible output operator $C$ such that $(C,A)$ is exactly observable, then $g(A)$ is bounded for all $g$'s with $g(-s) \in {\mathcal H}_{\infty}$, i.e., there exists a bounded ${\mathcal H}_{\infty}$-calculus. Moreover, we rediscover some well-known classes of generators also having a bounded ${\mathcal H}_{\infty}$-calculus.