Recently, physics-informed learning, a class of deep learning framework that incorporates the physics priors and the observational noise-perturbed data into the neural network models, has shown outstanding performances in learning physical principles with higher accuracy, faster training speed, and better generalization ability. Here, for the Hamiltonian mechanics and using the Koopman operator theory, we propose a typical physics-informed learning framework, named as \textbf{H}amiltonian \textbf{N}eural \textbf{K}oopman \textbf{O}perator (HNKO) to learn the corresponding Koopman operator automatically satisfying the conservation laws. We analytically investigate the dimension of the manifold induced by the orthogonal transformation, and use a modified auto-encoder to identify the nonlinear coordinate transformation that is required for approximating the Koopman operator. Taking the Kepler problem as an example, we demonstrate that the proposed HNKO in robustly learning the Hamiltonian dynamics outperforms the representative methods developed in the literature. Our results suggest that feeding the prior knowledge of the underlying system and the mathematical theory appropriately to the learning framework can reinforce the capability of the deep learning.