Equivalence of Hardy Submodules Generated by Polynomials

Abstract

In this paper, we obtain a complete classification under unitary equivalence for Hardy submodules on the polydisk which are generated by ideals of polynomials. Let I be an ideal of polynomials in n variables. Since I is generated by finitely many polynomials, I has a greatest common divisor p. So, I can be uniquely written as I=p L which is called the Beurling form of I. Let I1=p1L1, I2=p2L2. We prove that [I1] and [I2] are unitarily equivalent if and only if there are polynomials q1 and q2 with Z(q1)∩Dn=Z(q2)∩Dn=∅ such that |p1q1|=|p2q2| on Tn, and [p1L1]=[p1L2]. Consequently, two principal submodules [p1] and [p2] are unitarily equivalent if and only if there are polynomials q1 and q2 with Z(q1)∩Dn=Z(q2)∩Dn=∅ such that |p1q1|=|p2q2| on Tn. Furthermore, we give a complete similarity classification for submodules generated by homogeneous ideals. Finally, we point out that in the case of the Hardy module on the unit ball, [I1] and [I2] are unitarily equivalent if and only if they are equal. If I1 and I2 are homogeneous ideals, then [I1] and [I2] are quasi-similar if and only if I1=I2.