Let A be a continuous operator, having a continuous inverse and K be a compact operator: \(A,\quad K: \ell^ 2({\mathbb{Z}})\to \ell^ 2({\mathbb{Z}}).\) Suppose that the matrices A(i,j) and K(i,j), i,j\(\in {\mathbb{Z}}\), associated to A respectively to K are m-banded, i.e. there is an index \(\ell\) (the same for both matrices) so that \(A(i,j)=0\) and \(K(i,j)=0\) for \(j\not\in [i-\ell,\quad i-\ell +m].\) By generalized eigenvalue, respectively eigenvector we understand a scalar \(\mu\) \(\neq 0\) respectively a vector \(x\neq 0\), \(x\in \ell^ 2({\mathbb{Z}})\) so that \((\mu A-K)x=0.\) The authors prove that if x is a generalized eigenvector then \(| x(i)| \leq C\gamma^{| i|}\) for a suitable \(\gamma <1\) and \(C=C(\gamma,A,K)\). Analogous estimates are given for generalized eigenvalues and characteristic values. As an application, estimates for the characteristic values of the Mathieu differential equation \(y''+(a-2q \cos 2x)y=0\) are also obtained.