An ergodic theorem for classes of preconditioned matrices

We consider abstract classes of matrices {A} 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 A. 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 L-1 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. (C) 1998 Elsevier Science B.V. All rights reserved.