Inclusions between FK spaces and Kuttner's theorem

Source 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 ⊃ w 0 ( p ) then X ⊃ l ∞ . Here, w 0 ( p ) denotes the space of sequences strongly summable to 0, i.e. x ∈ w 0 ( p ) if and only if and l ∞ denotes the space of bounded sequences. Other proofs of Thorpe's extension and related results appear in Maddox[4, 5].