Ideals of square summable power series in several variables

Abstract

Let C ( z ) \mathcal {C}(z) be the Hilbert space of formal power series in z 1 , ⋯ , z r ( r ≧ 1 ) {z_1}, \cdots ,{z_r}(r \geqq 1) . An ideal of C ( z ) \mathcal {C}(z) is a vector subspace M \mathcal {M} of C ( z ) \mathcal {C}(z) which contains z 1 f ( z ) , ⋯ , z r f ( z ) {z_1}f(z), \cdots ,{z_r}f(z) whenever it contains f ( z ) f(z) . If B ( z ) B(z) is a formal power series such that B ( z ) f ( z ) B(z)f(z) belongs to C ( z ) \mathcal {C}(z) and | | B ( z ) f ( z ) | | = | | f ( z ) | | ||B(z)f(z)|| = ||f(z)|| , then the set M ( B ) \mathcal {M}(B) of all products B ( z ) f ( z ) B(z)f(z) is a closed ideal of C ( z ) \mathcal {C}(z) . In the case r = 1 r = 1 , Beurling showed that every closed ideal is of this form for some such B ( z ) B(z) . Here we give conditions under which a closed ideal is of the form M ( B ) \mathcal {M}(B) for r ≧ 2 r \geqq 2 .