Abstract
Results on first order Ext groups for Hilbert modules over the disk algebra are used to study certain backward shift invariant operator ranges, namely de Branges–Rovnyak spaces and a more general class called H(W;B) spaces. Necessary and sufficient conditions are given for the groups Ext1A(D)(H2C, H(W;B)) to vanish whereH2Cis thedualof the vector-valued Hardy module, H2C. One condition involves an extension problem for the Hankel operator with symbolB,ΓB, but viewed as a module map from H2Cinto H(W;B). The group Ext1A(D)(H2C, H(W;B))=(0) precisely whenΓBextends to a module map from L2Cinto H(W;B) and this in turn is equivalent to the injectivity of H(W;B) in the category of contractive HilbertA(D)-modules. This result applied to the de Branges–Rovnyak spaces yields a connection between the extension problem for the HankelΓB and the operator corona problem.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
