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Beurling type theorem on the Bergman space via the Hardy space of the bidisk. (English) Zbl 1191.47007

Let \(L_{a}^{2}\) be the Bergman space, i.e., the Hilbert space of the functions analytic in the unit disk of the complex plane satisfying the inequality \( \int_{|z|<1} |f(z)|^{2}\, \frac{dA(z)}{\pi}< 1\), where \(dA(z)\) is the area measure. Denote by \(M_z\) the Bergman shift, i.e., the multiplication operator on \(L_{a}^{2}\) induced by the identity function \(z\mapsto z\). A.Aleman, S.Richter and C.Sundberg [Acta Math.177, 275–310 (1996; Zbl 0886.30026)] proved a Beurling type theorem which says that every \(M_z\)-invariant subspace \(M \leq L_{a}^{2}\) is of the form \(M=[M\ominus zM]\), where \(M\ominus z M\) is the wandering subspace of \(M\) and \([M\ominus zM]\) denotes the smallest \(M_z\)-invariant subspace containing \(M\ominus zM\). In the present paper, a new proof of this theorem is given.

MSC:

47A15 Invariant subspaces of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
30H20 Bergman spaces and Fock spaces
46E20 Hilbert spaces of continuous, differentiable or analytic functions

Citations:

Zbl 0886.30026
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References:

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