Generalized Cauchy product and related operators on $\ell^p(\beta)$

In this paper first we give some necessary and sufficient conditions for the boundedness of the multiplication operator $D_f=M_{*\!\!\!\bigcirc,f}$ with respect to the generalized Cauchy product  $*\!\!\!\!\!\bigcirc$, on $\ell^p(\beta)$. Also, under certain conditions, we give the characterization of the extended eigenvalues and extended eigenvectors of the multiplication operator $M_{*\!\!\!\bigcirc,z}$ on $\ell^p(\beta)$. Finally we describe the commutants of $M_{*\!\!\!\bigcirc,z}$ and consequently the collection of all hyperinvariant subspaces of $M_{*\!\!\!\bigcirc,z}$.