  We propose a two-sided Lanczos method for the nonlinear eigenvalue problem
(NEP). This two-sided approach provides approximations to both the right and
left eigenvectors of the eigenvalues of interest. The method implicitly works
with matrices and vectors with infinite size, but because particular (starting)
vectors are used, all computations can be carried out efficiently with finite
matrices and vectors. We specifically introduce a new way to represent infinite
vectors that span the subspace corresponding to the conjugate transpose
operation for approximating the left eigenvectors. Furthermore, we show that
also in this infinite-dimensional interpretation the short recurrences inherent
to the Lanczos procedure offer an efficient algorithm regarding both the
computational cost and the storage.
