Can $f(Ax_k)-f(Ax^*)\le o(1/k)$ derives $f(x_k)-f(x^*)\le o(1/k)$?

Question

$\{x_k\}$ is a sequence which converges to $x^*\ (when\ k\to \infty)$, and $f$ is a convex and L-lipschitz smooth function, that means, for $\forall x,y\in dom(f),|\nabla f(x)-\nabla f(y)|\le L|x-y|$. We wonder is there such a matrix A, when we get the convergence rate as $f(Ax_k)-f(Ax^*)\le o(1/k)$, we can derive $f(x_k)-f(x^*)\le o(1/k)$ directly? And what the matrix looks like?