  Among the sets of sequences studied, difference sets of sequences are
probably the most common type of sets. This paper considers some $\ell_{p}$
type fractional difference sequence spaces via Euler gamma function. Although
we characterize compactness conditions on those spaces using the main tools of
Hausdorff measure of noncompactness, we can only obtain sufficient conditions
when the final space is $\ell _{\infty }$. However, we use some recent results
to exactly characterize the classes of compact matrix operators when the final
space is the set of bounded sequences.
