Extensions of pairs of linear transformations between infinite-dimensional vector spaces

Abstract

To a pair A, B: V→ W of linear maps between complex vector spaces attach the pair ( V, W) endowed with the operation (α, β)υ = (α A + β B)(υ), α,β ∈ C, υ ∈ V. A concept of rank, similar to the torsion-free rank of abelian groups, is definable for the systems ( V, W). With appropriate morphisms, the systems from an abelian category and Ext 1 can be construed as a vector space valued functor. We find all the cases in which Ext 1 (( V, W), ( X, Y)), with ( X, Y), ( V, W) indecomposable systems of rank 0 or 1, is finite-dimensional, and compute its dimension in these cases. This extends a former computation for finite-dimensional systems.