Multiplication operators on the weighted Sobolev disk algebra

Abstract

Let ⅅ be the unit disk in the complex plane ℂ. For α > −1, the weighted Sobolev disk algebra SA(ⅅ, dA_α ) consists of all analytic functions in the weighted Sobolev space W ^2,2(ⅅ, dA_α ). In this paper, we prove that the multiplication operator is similar to on SA(ⅅ, dA_α ) if and only if n = m, where n, m are positive integers. Then we characterize when a bounded operator P on SA(ⅅ, dA_α ) belongs to the commutant of by using the matrix representation of P. In addition, we compute the exact norm of M_z on SA(ⅅ, dA_α ). Finally, we prove that on the unweighted Sobolev disk algebra SA(ⅅ) the restrictions of to different invariant subspaces z^kSA(ⅅ) (k ≥ 1) are not unitarily equivalent, and the restrictions of (n ≥ 2) to different invariant subspaces S_j (0 ≤ j < n) are also not unitarily equivalent.