G- Cyclicity And Somewhere Dense Orbit

let H be an infinite – dimensional separable complex Hilbert space, and S be a multiplication semigroup of C with 1. An operator T is called G-cyclic over S if there is a non-zero vector x ∈ H such that {αT^n x α ∈|S, n ≥0} is norm-dense in H. Bourdon and Feldman have proved that the existence of somewhere dense orbits implies hypercyclicity. We show the corresponding result for G-cyclicity