It is natural to approach the Cauchy problem for the Hamilton-Jacobi equation \(u_ t+H(t,x,u,Du)=0\) in \((0,T)\times {\mathbb{R}}^ n\) with \(u(0,x)=u_ 0(x)\) with the theory of nonlinear semigroups. This approach and its connection with Crandall and Lions viscosity solutions was introduced also by Crandall and Lions. However, many important generalizations of the original Crandall and Lions theory of viscosity solutions, such as allowing discontinuous Hamiltonians due to \textit{H. Ishii} [``Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets'', Bull. Fac. Sci. Eng., Chuo Univ., Ser. I 28, 33-77 (1985)], have not been studied using the evolution operator approach. The present paper, using that approach, constructs an operator which gives a viscosity solution with the Ishii hypotheses (for bounded solutions).