Projective generators inHardy and Bergman spaces

✍ Scribed by B. Korenblum; T.L. Lance; M.I. Stessin

• Publisher: Elsevier Science
• Year: 2000
• Tongue: French
• Weight: 76 KB
• Volume: 124
• Category: Article
• ISSN: 0007-4497
• DOI: 10.1016/s0007-4497(00)01058-7

✦ Synopsis

One form of the celebrated theorem of Beurling states that if M is a z-invariant subspace of the Hardy space H 2 and P M is the orthogonal projection from H 2 onto M, then g = P M ( 1) is either a generator of M (that is M = gC[z]) or g = 0. In the latter case there is an n such that P M (z n ) generates M. If M is a z-invariant subspace of the Bergman space A 2 which has the codimension 1 property (or, equivalently, is singly generated), the same result holds. This was proved in the recent paper [1]. Furthermore, the Bergman kernel centered at any point of the unit disk satisfies the same property for A 2 (and the Cauchy kernel for the H 2 setting). Namely, the A 2 -orthogonal projection of a kernel function into any singly generated z-invariant subspace is either a generator for this subspace, or identically equal to zero. This can be deduced from the above by a change of variables. Below we write the explicit expression of this projection in terms of weighted reproducing kernels.

We call a function which satisfies the above property a projective generator. This paper, which grew out of the authors' attempt to better understand factorization in the Bergman space, addresses the question of describing all projective generators of A 2 (and H 2 ). One possible generalization of this question is the following. Given a Bergman