On the Carleson measure criterion in linear systems theory

TL;DR

This paper extends Carleson measure criteria for admissibility in linear systems from Hilbert spaces to more general Banach spaces, providing new results for analytic semigroups and reciprocal systems.

Contribution

It generalizes existing Carleson measure criteria from $L^2$-admissibility to $L^p$-admissibility on $ ext{ell}_q$ spaces and introduces a novel approach for analytic diagonal semigroups.

Findings

Extended Carleson measure criteria to $L^p$-admissibility on $ ext{ell}_q$ spaces.

Presented a new criterion for analytic diagonal semigroups independent of Laplace transforms.

Compared criteria to establish $L^p$-admissibility for reciprocal systems.

Abstract

In Ho, Russell, and Weiss, a Carleson measure criterion for admissibility of one-dimensional input elements with respect to diagonal semigroups is given. We extend their results from the Hilbert space situation $X=\ell_2$ and $L^2$--admissibility to the more general situation of $L^p$--admissibility on $\ell_q$--spaces. In case of analytic diagonal semigroups we present a new result that does not rely on Laplace transform methods. A comparison of both criteria leads to result of $L^p$--admissibility for reciprocal systems in the sense of Curtain.