Matrix algebra preconditioners for multilevel Toeplitz systems do not insure optimal convergence rate

In the last decades several matrix algebra optimal and superlinear preconditioners (those assuring a strong clustering at the unity) have been proposed for the solution of polynomially ill-conditioned Toeplitz linear systems. The corresponding generalizations for multilevel structures are neither optimal nor superlinear (see e.g. Contemp. Math. 281 (2001) 193). Concerning the notion of superlinearity, it has been recently shown that the proper clustering cannot be obtained in general (see Linear Algebra Appl. 343-344 (2002) 303; SIAM J. Matrix Anal. Appl. 22(1) (1999) 431; Math. Comput. 72 (2003) 1305). In this paper, by exploiting a proof technique previously proposed by the authors (see Contemp. Math. 323 (2003) 313), we prove that the spectral equivalence and the essential spectral equivalence (up to a constant number of diverging eigenvalues) are impossible too. In conclusion, optimal matrix algebra preconditioners in the multilevel setting simply do not exist in general and therefore the search for optimal iterative solvers should be oriented to different directions with special attention to multilevel/multigrid techniques.