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, dAα ) consists of all analytic functions in the weighted Sobolev space W 2,2(ⅅ, dAα ). In this paper, we prove that the multiplication operator is similar to on SA(ⅅ, dAα ) if and only if n = m, where n, m are positive integers. Then we characterize when a bounded operator P on SA(ⅅ, dAα ) belongs to the commutant of by using the matrix representation of P. In addition, we compute the exact norm of Mz on SA(ⅅ, dAα ). Finally, we prove that on the unweighted Sobolev disk algebra SA(ⅅ) the restrictions of to different invariant subspaces zkSA(ⅅ) (k ≥ 1) are not unitarily equivalent, and the restrictions of (n ≥ 2) to different invariant subspaces Sj (0 ≤ j < n) are also not unitarily equivalent.