The importance of the block or band Lanczos method for many-body Green’s function calculations of atomic and molecular systems is discussed. The usual computation schemes for determining the Green’s function involve the diagonalization of Hermitian secular matrices. Considerable numerical difficulties arise, on the one hand, from the size of these matrices and, on the other hand, from the large number of eigenvalues and eigenvectors which often need to be computed in practice. In the case of the one-particle Green’s function it is shown how the computational effort of the diagonalization process can be substantially reduced using block Lanczos. The proposed procedure which consists of a block Lanczos "prediagonalization" and a subsequent diagonalization of the resulting smaller secular matrices quite naturally exploits the specific structure of the secular problems encountered. Its computational performance is demonstrated in a model application to the benzene molecule. The calculation of the complete valence-shell ionization spectra of the systems BeF42-, BeF3-, and BeF2 is devised as a further application of the method in the particular case where the treatment of the full secular problem is computationally prohibitively expensive.