The present paper develops a theme outlined in a previous article, ibid. 654-678 (1986; review above). A calculus is defined for a class of Pettis integrals of operator valued functions, turning it into an algebra of operators on \(L^ p({\mathbb{R}}^ d)\). The symbol of the operators differs from the usual definition, in that the Fourier transform is used with respect to the space variables, and the Laplace transform is used for the time variable. The main result is that whenever the Pettis integral operators are invertible in the algebra defined with the pseudo-product, they are asymptotically invertible with respect to the product of the composition of operators. The result is applied to the construction of the resolvent of an elliptic partial differential operator on \(L^ p({\mathbb{R}}^ d)\). It is shown that when the partial differential operator is represented as the product of the analogous operator for the Laplacian, and a perturbation, then it turns out that the perturbation is a Pettis integral operator, which is invertible with respect to the pseudo-product algebra. It then follows that the resolvent of the initial operator is the product of the resolvent of the Laplacian, and the inverse of the perturbation. This representation of the resolvent of the elliptic differential operator has recently been used in the construction of diffusion semigroups on \({\mathbb{R}}^ d\) [''Semigroups and diffusion processes'', Math. Proc. Cambridge Phil. Soc., to appear].