Equivalence of Hardy Submodules Generated by Polynomials

✍ Scribed by Kunyu Guo

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 206 KB
• Volume: 178
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3659

✦ Synopsis

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.

We prove that [I 1 ] and [I 2 ] are unitarily equivalent if and only if there are polynomials q 1 and q 2 with Z(q

Consequently, two principal submodules [ p 1 ] and [ p 2 ] are unitarily equivalent if and only if there are polynomials q 1 and q 2 with Z(q 1 ) & D n =Z(q 2 ) & D n =< such that | p 1 q 1 |= | p 2 q 2 | on T n . 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, [I 1 ] and [I 2 ] are unitarily equivalent if and only if they are equal. If I 1 and I 2 are homogeneous ideals, then [I 1 ] and [I 2 ] are quasi-similar if and only if I 1 =I 2 .