Ideals of square summable power series

Abstract

The closed invariant subspaces of multiplication by z in H2 were determined by Beurling [1, Theorem IV, p. 253]. Vector generalizations of this theorem are known (Halmos [3] and the author [6]), but they involve an unnecessary use of analysis. We can now prove the theorem of [6] by purely algebraic and geometric methods. To emphasize these methods, we work with sequences, which we write as formal power series, rather than functions analytic in the unit disk. Let e be a Hilbert space with elements denoted by a, b, c, and with norm j * |. If b is a vector in C, then b is the linear functional on C such that ba = (a, b) for every a in C. A formal power series is a sequence (ao, a,, a2, * * ) written f(z) = Janzn with an indeterminate z. Let f(z) = 2anZn and g(z) = EbnZn be formal power series with coefficients an and bn in C; let B(z) = EBnzn be a formal power series whose coefficients Bn are (bounded) operators in C; let a be a complex number, and let c be a vector in C. Then f(z) +g(z), af(z), zf(z), and B(z)f(z) are the formal power series Z(an +bn)zn, (aan)Zn, (jan)zn, and ( X_0 Bkan-k)Zn, respectively. A sequence (fk(z)) of formal power series with coefficients in C is said to be formally convergent if, for each n = 0, 1, 2, * * * , the corresponding sequence of nth coefficients is convergent. Let C(z) be the Hilbert space of formal power series f(z) = anZn with coefficients an in C, such that