In a recent paper, one of the authors -- along with co-authors -- 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 z². In recent times, several researchers have replaced z² by an arbitrary inner function I and this has proved important and fruitful in applications such as to interpolation problems of the Pick-Nevanlinna 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.

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)/ 2012-EMR-I)