Compact operators into the spaces of strongly C1C1 summable and bounded sequences

We establish the necessary and sufficient conditions for the entries of an infinite matrix to map any of the classical sequence spaces ℓpℓp(1≤p≤∞)(1≤p≤∞), c0c0 and cc into the spaces w0w0, ww and w∞w∞ of all sequences that are strongly summable to zero, strongly summable and strongly bounded, by the Cesàro method of order 1. We also give the representations of the general bounded linear operators from cc into any of the spaces w0w0, ww and w∞w∞, and compute or estimate the Hausdorff measure of noncompactness in each case. Finally, we apply our results to characterise the compact linear operators between the spaces mentioned above, and generalise two classical results by Steinhaus (1911) [5], Maddox (1967) [31], and Cohen and Dunford (1937) [24].