2014/10/11 by Sophie Grivaux, Grivaux, Sophie
Mathematics · #Advanced Harmonic Analysis Research #advanced mathematical theories #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.1410.2957
A bounded operator on a real or complex separable infinite-dimensional Banach\nspace Z is universal in the sense of Glasner and Weiss if for every\ninvertible ergodic measure-preserving transformation T of a standard Lebesgue\nprobability space (X, mathcal B,\μ ), there exists an A-invariant\nprobability measure \ν on H with full support such that the two dynamical\nsystems (X, mathcal B,\μ ;T) and (H, mathcal BH,\ν ;A) are\nisomorphic. We present a general and simple criterion for an operator to be\nuniversal, which allows us to characterize universal operators among unilateral\nor bilateral weighted shifts on \ℓp or c0, show the existence of\nuniversal operators on a large class of Banach spaces, and give a criterion for\nuniversality in terms of unimodular eigenvectors. We also obtain similar\nresults for operators which are universal for all ergodic systems (not only for\ninvertible ones), and study necessary conditions for an operator on a Hilbert\nspace to be universal.\n