Reproducing Kernel Hilbert Spaces and entropy Kolmogorov numbers on compact Lie Groups

On a compact Lie group $G$, we consider the reproducing kernel Hilbert space ${\mathcal{H}}_K$ associated with the integral kernel $K$ of a left-invariant, positive, symmetric, trace class integral operator on $L^2(G)$. We present lower and upper asymptotic estimates for the entropy Kolmogorov numbers (also called covering numbers) for the embedding of ${\mathcal{H}}_K$ into the space $C(G)$ of continuous functions on $G$.