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2010 | 200 | 3 | 221-246
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Spectral theory and operator ergodic theory on super-reflexive Banach spaces

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On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that
$sup_{n∈ ℕ, z∈ 𝕋} || ∑_{0<|k|≤n} (1 - |k|/(n+1))k^{-1}z^{k}U^{k}|| < ∞$. (*)
Trigonometrically well-bounded operators permeate many settings of modern analysis, and this note highlights the advances in both their spectral theory and operator ergodic theory made possible by a recent rekindling of interest in the R. C. James inequalities for super-reflexive spaces. When the James inequalities are combined with Young-Stieltjes integration for the spaces $V_{p}(𝕋)$ of functions having bounded p-variation, it transpires that every trigonometrically well-bounded operator on a super-reflexive space X has a norm-continuous $V_{p}(𝕋)$-functional calculus for a range of values of p > 1, and we investigate the ways this outcome logically simplifies and simultaneously expands the structure theory, Fourier analysis, and operator ergodic theory of trigonometrically well-bounded operators on X. In particular, on a super-reflexive space X (but not on a general relexive space) a theorem of Tauberian type holds: the (C,1) averages in (*) corresponding to a trigonometrically well-bounded operator U can be replaced by the set of all the rotated ergodic Hilbert averages of U, which, in fact, is a precompact set relative to the strong operator topology. This circle of ideas is facilitated by the development of a convergence theorem for nets of spectral integrals of $V_{p}(𝕋)$-functions. In the Hilbert space setting we apply the foregoing to the operator-weighted shifts which are known to provide a universal model for trigonometrically well-bounded operators on Hilbert space.
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  • Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, U.S.A.
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