Let $cal{H}$ be a separable, infinite dimensional, complex Hilbert space. We call "an operator $cal{T}$ acting on $cal{H}$ has a star moment sequence supported on a set K" when there exist nonzero vectors u and v in $cal{H}$ and a positive Borel measure ${mu}$ such that <$T^{*j}T^ku$, v> = ${^intlimits_{K}};{{ar{z}}^j};{{ar{z}}^k};dmu$ for all j, $k;geq;0$. We obtain a characterization to find a representing star moment measure and discuss some related properties.