On the Carleson measure criterion in linear systems theory

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 $X={\mathrm{\ell }}_2$ and $L^2$–admissibility to the more general situation of $L^p$–admissibility on ${\mathrm{\ell }}_q$–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.