Multiplication operators on the weighted Sobolev disk algebra 1

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 M_zn is similar to M_zm 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 A′ (M_zn ) of M_zn 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 M_zn to different invariant subspaces z^kSA(D) (k ≥ 1) are not unitarily equivalent, and the restrictions of M_zn (n ≥ 2) to different invariant subspaces S_j (0 ≤ j < n) are also not unitarily equivalent.