ZERO-BASED INVARIANT SUBSPACES IN THE BERGMAN SPACE

Authors : Bouabdallah FATIHA, Bendaoud ZOHRA

Pages : 277-285

Doi:10.1501/Commua1_0000000849

View : 7 | Download : 8

Publication Date : 2018-02-01

Article Type : Research Paper

Abstract :It is known that Beurling’s theorem concerning invariant subspaces is not true in the Bergman space insert ignore into journalissuearticles values(in contrast to the Hardy space case);.However, Aleman, Richter, and Sundberge proved that every cyclic invariantasubspace in the Bergman space Lpinsert ignore into journalissuearticles values(D);, 0 < p < +1, is generated by its extremal function. This implies, in particular, that for every zero-based invariantsubspace in the Bergman space the Beurling’s theorem stands true. Here, wecalculate the reproducing kernel of the zero-based invariant subspace Mninathe Bergman space L2insert ignore into journalissuearticles values(D); where the associated wandering subspace Mnis one-dimensional, and spanned by the unit vector Gninsert ignore into journalissuearticles values(z); =zMn p n + 1zn
Keywords : Bergman space, inner function, Beurlings theorem, kernel function