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 W 2 , 2 ( D ) . In this note, we prove that M z n is similar to ⨁ 1 n M z on SA ( D ) . Then using the matrix manipulations combined with operator theory methods, we characterize the reducing subspaces of M z n on SA ( D ) .