This study aims to construct the BK-spaces ${\ell }_p(\mathcal{M})$ and ${\ell }_{\mathrm{\infty }}(\mathcal{M})$ as the domains of the conservative Motzkin matrix $\mathcal{M}$ obtained by using Motzkin numbers. It investigates topological properties, obtains Schauder basis, and then gives inclusion relations. Additionally, it expresses $\alpha$-, $\beta$-, and $\gamma$-duals of these spaces and submits the necessary and sufficient conditions of the matrix classes between the described spaces and the classical spaces. In the last part, the characterization of certain compact operators is given with the aid of the Hausdorff measure of non-compactness.