In this paper, we study Diriehlet operators on certain smooth Banach spaces. We establish the well-known Kato's inequality in our general infinite dimensional setting. By applying this,we show the essential self-adjointness of Dirichlet operators with non-constant diffusion part on certain smooth Banach spaces. We also provide an approximation criterion for essential self-adjointness of Dirichlet operators with identity diffusion part on M-type 2 Banach spaces via the classical notion of semi inner-product.