Projective generators in Hardy and Bergman spaces (Q1591583)

A function \(f\) in the Bergman space \(A^2\) (respectively, the Hardy space \(H^2)\) is called a projective generator of defect \(n\) if \[ \text{codim}_{[g]} [P_{[g]}f]\leq n \] for any \(g\in A^2\) (respectively, \(H^2)\) such that \(P_{[g]}f\neq 0\), and if equality holds for some \(g\). Here \(P_{[g]}\) stands for the orthogonal projection onto the minimal closed \(z\)-invariant subspace \([g]\) of \(A^2\) (respectively, \(H^2)\) containing \(g\). It is known from the theorem of Beurling and its generalization to Bergman spaces that \(f=1\) is a projective generator of order zero and the authors ask for a description of all projective generators. A necessary (but not sufficient) condition for a function \(f\in A^2\) (respectively, \(H^2)\) to be a generator of order \(n\) is that \(f\) is a linear combination of at most \(n+1\) Bergman kernels (respectively, Cauchy kernels). A necessary and sufficient condition for a function \(f\in A^2\) (respectively, \(H^2)\) to be a generator of order 0 is that \(f\) is a multiple of a Bergman kernel (respectively, Cauchy kernel). The most interesting step in the proof of these fact is the use of the Borsuk-Ulam theorem that states that an odd continuous map \(f:\mathbb{S}^n\to \mathbb{R}^m\) \((1\leq n,m\leq n)\) on the sphere vanishes for some \(x\in\mathbb{S}^n\).