Tauberian pairs of closed subspaces of a Banach space

Abstract We introduce the notions of tauberian, cotauberian and weakly compact pair of closed subspaces of a Banach space. The theory produced by these notions is richer than that of the corresponding operators since an operator can be regarded as a suitable pair of closed subspaces. We investigate into these classes of pairs of subspaces and describe several applications to define some notions of indecomposability for Banach spaces and to extend definitions from the case of bounded operators to the case of closed operators.