Operator ranges and spaceability

Abstract

Recent contributions on spaceability have overlooked the applicability of results on operator range subspaces of Banach spaces or Frechet spaces. Here we consider general results on spaceability of the complement of an operator range, some of which we extend to the complement of a union of countable chains of operator ranges. Applications we give include spaceability of the non-absolutely convergent power series in the disk algebra and of the non absolutely p-summing operators between certain pairs of Banach spaces. Another application is to ascent and descent of countably generated sets of continuous linear operators, where we establish some closed range properties of sets with finite ascent and descent.