 

Ajay Kumar, Niteesh Sahni, and Dinesh Singh
Invariance under finite Blaschke factors on BMOA view print


Published: 
November 13, 2017 
Keywords: 
Blaschke factor, Hardy space, H^{1}BMOA duality, invariant subspace, backward shift 
Subject: 
Primary 47B37; Secondary 47A25 


Abstract
This paper describes completely the invariant subspaces of the operator of multiplication by a finite Blaschke factor on the Banach space BMOA of analytic functions with bounded mean oscillation on the unit circle in the complex plane. As a simple application, we describe by very elementary means, the invariant subspaces of the coanalytic Toeplitz operator T_{\overline{B}} on H^{1}. In the simplest case when B(z) = z, the invariant subspaces of T_{\overline{B}} on H^{1} were described by fairly deep arguments until the appearance of an elementary proof by two of the authors (Sahni & Singh). In recent times, the common invariant subspaces of the operators of multiplication by B^{2} and B^{3}, first in the case of z^{2} and z^{3}, and then for an arbitrary finite Blaschke B, have proved to be critical in the context of NevanlinnaPick type interpolation on H^{2}. Thus, keeping in mind the importance of invariant subspaces, we also offer a characterization of the common invariant subspaces of these operators on BMOA. Our proofs are that much more technical. Again, as an application, we obtain the common invariant subspaces of T_{\overline{B2}} and T_{\overline{B3}} on the Hardy space H^{1}.


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

