We establish the characterisations of the classes of bounded linear operators from the generalised Hahn sequence space h d , where d is an unbounded monotone increasing sequence of positive real numbers, into the spaces w 0 , w and w ∞ of sequences that are strongly summable to zero, strongly summable and strongly bounded by the Cesàro method of order one. Furthermore, we prove estimates for the Hausdorff measure of noncompactness of bounded linear operators from h d into w, and identities for the Hausdorff measure of noncompactness of bounded linear operators from h d to w 0 , and use these results to characterise the classes of compact operators from h d to w and w 0. Finally, we provide an example for an application of our results