Das, Suman (2018) *Beurling's Theorem for the Bergman space.* Masters thesis, Indian Institute of Science Education and Research Kolkata.

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## Abstract

It is interesting to study Hilbert spaces of analytic functions on the unit disk D and the operators on them. One of the most elementary of these operators is multiplication by the coordinate function z. The Hardy space H² constructs an important example in this area. A famous theorem by A. Beurling characterizes the invariant subspaces of H², and thus the invariant subspaces of the unilateral shift. In this thesis, we will be interested in the Bergman space L²a. It has been known for some time that the invariant subspace lattice of L2 a is very complicated when compared to that of H². In particular, not every invariant subspace of L²a is cyclic. Nevertheless, the following analogue of Beurling's Theorem is true and is the main result of this thesis: Theorem. Let M be an invariant subspace of L²a. Then: M = [M Ɵ zM]. Thus, as in the Hardy space case, invariant subspaces of L²a are in a one-to-one correspondence with their wandering subspaces.

Item Type: | Thesis (Masters) |
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Additional Information: | Supervisor: Dr. Subrata Shyam Roy |

Uncontrolled Keywords: | Bergman Space; Beurling's Theorem; Wandering Subspace Theorem |

Subjects: | Q Science > QA Mathematics |

Divisions: | Department of Mathematics and Statistics |

Depositing User: | IISER Kolkata Librarian |

Date Deposited: | 27 Nov 2018 11:06 |

Last Modified: | 27 Nov 2018 11:07 |

URI: | http://eprints.iiserkol.ac.in/id/eprint/693 |

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