 

Ajay Kumar, Niteesh Sahni, and Dinesh Singh
Invariance under bounded analytic functions: generalizing shifts view print


Published: 
October 29, 2016 
Keywords: 
Invariant subspace, inner function, uniform algebra, compact abelian group, multiplier algebra of BMOA 
Subject: 
Primary 47B37; Secondary 47A25 


Abstract
In a recent paper, one of the authors  along with coauthors  extended the famous theorem of Beurling to the context of subspaces that are invariant under the class of subalgebras of H^{∞} of the form IH^{∞}, where I is the inner function z^{2}. In recent times, several researchers have replaced z^{2} by an arbitrary inner function I and this has proved important and fruitful in applications such as to interpolation problems of the PickNevanlinna type. Keeping in mind the great deal of interest in such problems, in this paper, we provide analogues of the above mentioned IH^{∞} related extension of Beurling's theorem in the setting of the Banach space BMOA, in the context of uniform algebras, on compact abelian groups with ordered duals and the Lebesgue space on the real line. We also provide a significant simplification of the proof of Beurling's theorem in the setting of uniform algebras and a new proof of Helson's generalization of Beurling's theorem in the context of compact abelian groups with ordered duals.


Acknowledgements
The research of the first author is supported by the Junior Research Fellowship of the Council of Scientific and Industrial Research, India (Grant no. 09/045(1232)/ 2012EMRI)


Author information
Ajay Kumar:
Department of Mathematics, University of Delhi, Delhi (India) 110007
nbkdev@gmail.com
Niteesh Sahni:
Department of Mathematics, Shiv Nadar University, Dadri, Uttar Pradesh (India) 201314
niteeshsahni@gmail.com
Dinesh Singh:
Department of Mathematics, University of Delhi, Delhi (India) 110007
dineshsingh1@gmail.com

