Backward Shift Invariant Operator Ranges |
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Authors: | Sarah H Ferguson |
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Institution: | Department of Mathematics, Purdue University, Lafayette, Indiana, 47907-1395 |
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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 (W; B) spaces. Necessary and sufficient conditions are given for the groups Ext1A()(, (W; B)) to vanish whereis thedualof the vector-valued Hardy module, H2. One condition involves an extension problem for the Hankel operator with symbolB,ΓB, but viewed as a module map from H2into (W; B). The group Ext1A()(, (W; B))=(0) precisely whenΓBextends to a module map from L2into (W; B) and this in turn is equivalent to the injectivity of (W; B) in the category of contractive HilbertA()-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. |
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