Inclusions between FK spaces and Kuttner's theorem

Abstract

The result known as Kuttner's theorem [2] asserts that if 0 < p < 1 and A is a Toeplitz matrix then there is a sequence which is strongly Cesàro summable with index p but which is not A summable. This theorem was extended by Maddox[3] to coregular matrices, and Thorpe [9] gave a further extension by showing that if 0 < p < 1 and X is a locally convex FK space with X ⊃ w0(p) then X ⊃ l∞. Here, w0(p) denotes the space of sequences strongly summable to 0, i.e. x∈w0(p) if and only ifand l∞ denotes the space of bounded sequences. Other proofs of Thorpe's extension and related results appear in Maddox[4, 5].