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An invariant supspace of the Bergman space having the codimension two property. (English) Zbl 0783.30040

A constructive example is given of a \(z\)-invariant subspace \(J\) of the Bergman space \(L^ 2_ a(\mathbb{D})\) such that \(zJ\) has codimension 2 in \(J\). The construction, which is based on Kristian Seip’s characterization of sampling and interpolating sequences for the space \(L^ 2_ a(\mathbb{D})\), is then generalized to obtain, for each integer \(n\geq 2\), a \(z\)-invariant subspaces \(J\) such that \(zJ\) has codimension \(n\) in \(J\).

MSC:

30H05 Spaces of bounded analytic functions of one complex variable
47B99 Special classes of linear operators
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
46C99 Inner product spaces and their generalizations, Hilbert spaces
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