On similarity and reducing subspaces of multiplication operator on Sobolev disk algebra

Let D be the unit disk and SA(D) denote the Sobolev disk algebra which consists of all analytic functions in the Sobolev space W2,2(D). In this note, we prove that Mzn is similar to ⨁1nMz on SA(D). Then using the matrix manipulations combined with operator theory methods, we characterize the reducing subspaces of Mzn on SA(D).