An admissible observation operator is zero-class admissible if the norm of the output map
tends to zero as the time tends to zero. Sufficient and necessary conditions for zero-class
admissibility of observation operators are developed and a modified Weiss condition is
studied. It is shown that the modified Weiss condition is in general necessary, but not
sufficient for zero-class admissibility. For several important classes of C0-semigroups it is
proved that the modified Weiss condition is indeed equivalent to zero-class admissibility …