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 Ext 1 A(D) (H 2 C , H(W; B)) to vanish where H 2 C is the dual of the vector-valued Hardy module, H 2 C . One condition involves an extension problem for the Hankel operator with symbol B, 1 B , but viewed as a module map from H 2 C into H(W ; B). The group Ext 1 A
C into H(W ; B) and this in turn is equivalent to the injectivity of H(W ; B) in the category of contractive Hilbert A(D)-modules. This result applied to the de Branges Rovnyak spaces yields a connection between the extension problem for the Hankel 1 B and the operator corona problem. 1997 Academic Press 0. INTRODUCTION Cohomological techniques are used to study certain backward shift invariant operator ranges contained in vector-valued Hardy space. These operator ranges can be viewed as contractive modules over the disk algebra, A(D), and contain as a proper subclass the de Branges Rovnyak spaces introduced in [dBR1]. The main problem considered here is that of determining when these modules are injective in the contractive category of Hilbert A(D)-modules. The spaces introduced here, called H(W ; B) spaces, are operator ranges determined by a positive Toeplitz operator and a vectorial Hankel operator. Specifically, let D be a separable Hilbert space and let H 2 D denote the usual Hardy class of D-valued functions on the the unit circle. Let W # L (L(D)) be a positive operator-valued function on the unit circle and let 1 B : H 2 C Γ H 2 D be a Hankel operator; i.e. 1 B S C =S* D 1 B where S D is the unilateral shift on H 2 D . As a Hilbert space H(W ; B) is the range of the row operator (T 1Γ2 W , 1 B ) equipped with the quotient norm, called the article no.