Abstract: In this paper, by using operator theory in Banach space, we mainly study spectral approximation theory for the case of an inhomogeneous, anisotropically scattering, and arbitrarily finite convex body with any cavity in an L^p context for generall p,1 ≤ p<∞. We prove that solutions of the time-dependent multigroup systems converge, in a suitable sense, to the corresponding solution of the exact time-dependent transport equation; the eigenvalues and the corresponding eigenfunctions and generalized eigenfunctions of the exact transport operator are approximated respectively by the eigenvalues and the corresponding eigenfunctions and generalized eigenfunctions of the multigroup transport operotors; the dominant eigenvalues and the corresponding positive eigenfunctions of the multigroup transport operators (the existence is proved, too) accumulate to those of the exact transport operators; the orders of magnitude of the rate of convergence are given respectively. Other results are obtained to compare the dimensions of eigenspaces and generalized eigenspaces of the transport operator with those of the multigroup transport operators.