Let 𝒜 be a Banach algebra and let J be a closed ideal of 𝒜 such that φ|J≠0 for some nonzero character φ on 𝒜. We obtain some relations between the existence of compact and weakly compact multipliers on J and on 𝒜 in some sense. Then we apply these results to hypergroup algebra L1(K) when K is a locally compact hypergroup. In particular, for a closed ideal J in L1(K) we prove that K is compact if and only if there is f∈J such that φ1(f)≠0 and the multiplication operator λf:g↦g∗f is weakly compact on J. Using this, we study Arens regularity of J whenever it has a bounded left approximate identity. Finally, we apply these results on some abstract Segal algebras with respect to the L1(K).