Remark on Co-Null Matrices

Extract

A well-known theorem of Copping [2] states that a conservative matrix with a bounded left inverse cannot evaluate a bounded divergent sequence. (Definitions are given in the next paragraph.) A proof was given by Parameswaran [3, Theorem 6. 1], using only the simplest Banach-space ideas. This proof, however, is valid only for co-regular methods; it was stated in [3, Theorem 6. 2] that a co-null matrix cannot have a bounded left inverse, but the proof there given is incorrect, as it uses for co-null methods a theorem established only for co-regular. It would be desirable to have a short independent proof of this known result, which excludes co-null matrices from consideration in Copping' s theorem. This is furnished by the slightly more general result given below.