Examples of Discrete Operators with a Pure Point Spectrum of Finite Multiplicity

One constructs operators acting on l2Zm‘ (or l2Zm‘p), m;p 1, with a real pure point spectrum of finite multiplicity by perturbing diagonal matrices using a KAM procedure. The point spectrum can be dense on an interval or a Cantor set of measure zero. The basic fact here is to remark that for perturbations built up with an infinite number of block diagonals, regularly separated, it is possible to deal with eigenvalues of multiplicity strictly greater than 1. Examples of discrete operators associated with discretization of systems of partial differential equations are given