Let $E$ be a locally convex Hausdorff space with continuous dual $E'$ and sequentially continuous dual $E^s$. In this paper, we show that if $E$ is an $\mathcal A$-space, then $(E,\sigma(E,E^s))$, $(E,\beta(E,E^s))$, $(E^s,\sigma(E^s,E))$ and $(E^s,\beta(E^s,E))$ are all $\mathcal A$-spaces. In particular, if $E$ is a Mazur $\mathcal A$-space, then $(E',\sigma(E',E))$ and $(E',\beta(E',E))$ are both $\mathcal A$-spaces. We apply the obtained results to generalize the adjoint Theorem on operators with the domain being a locally convex $\mathcal A$-space.