Beurling's Theorem for the Bergman space

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

PDF (MS dissertation of Suman Das (13MS019)) 13MS019.pdf - Submitted Version Restricted to Repository staff only Download (608kB)

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.

Actions (login required)

View Item