Many sequence spaces arise from different concepts of summability. Recent results obtained by Altay, Ba\c{s}ar and Malkowsky [2] are related to strong Ces\`{a}ro summability and boundedness. They determined $\beta$-duals of the new sequence spaces and characterized some classes of matrix transformations on them. Here, we will present new results supplementing their research with the characterization of classes of compact operators on those spaces.