Backward Shift Invariant Operator Ranges

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.