An ergodic theorem for classes of preconditioned matrices Original Research Article

We consider abstract classes of matrices {image} satisfying some structural conditions and, in particular, satisfying a crucial assumption about the asymptotic distribution of eigenvalues. We prove a similar distribution property for classes of preconditioned matrices constructed by using representants of {image}. As a particular case, this result applies to preconditioned matrices coming from several important contexts: Finite Differences and Faedo-Ritz-Galerkin linear systems associated with elliptic and semielliptic boundary value problems, very general Hermitian Toeplitz structures generated by multivariate Ll functions. This result answers in the positive some structural questions raised by Tyrtyshnikov [E. Tyrtyshnikov, Linear Algebra Appl. 207 (1994) 225–249] and by the author [S. Serra, Linear Algebra Appl. 267 (1997) 139–161; S. Serra, SIAM J. Numer. Anal., in press] in the Toeplitz context.