Domain of the Cesàro mean of order α in Maddox’s space ℓ(p)

Abstract

The sequence space ?(p) was defined by I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford (2), 18 (1967), 345-355. Here, we introduce the paranormed Cesaro sequence space ?(C?, p) of order ?, of non-absolute type as the domain of Cesaro mean C? of order ? and prove that the spaces ?(C?, p) and ?(p) are linearly paranorm isomorphic. Besides this, we compute the ?-, ?- and ?-duals of the space ?(C?, p) and construct the basis of the space ?(C?, p) together with the characterization of the classes of matrix transformations from the space ?(C?, p) into the spaces ?? of bounded sequences and f of almost convergent sequences, and any given sequence space Y , and from a given sequence space Y into the sequence space ?(C?, p). Finally, we emphasize on some geometric properties of the space ?(C?, p).