Let K be a linear compact operator in a Banach space. The eigenvalues and eigenelements of K are to be approximated. Let \(\tilde K\) be an operator of a finite rank such that \(r(\Delta)<| \lambda_ 0|\) where r(\(\Delta)\) is the spectral radius of \(\Delta =K-\tilde K\) and \(\lambda_ 0\in \sigma (K)\) is the eigenvalue under approximation. The authors propose to determine the approximate eigenvalues from the equation \[ (\lambda^ nI-\sum^{n-1}_{j=0}\lambda^{n-1-j}\Delta^ j\tilde K)y=0. \] They prove the corresponding convergence theorems and discuss the practical aspects of the method.