  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.
