### Rotation methods in operator ergodic theory.

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On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that $su{p}_{n\in \mathbb{N},z\in}\left|\right|{\sum}_{0<\left|k\right|\le n}(1-|k|/(n+1)){k}^{-1}{z}^{k}{U}^{k}\left|\right|<\infty $. (*) 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...

Let $f\in {V}_{r}\left(\right){\cup}_{r}\left(\right)$, where, for 1 ≤ r < ∞, ${V}_{r}\left(\right)$ (resp., ${}_{r}\left(\right)$) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values, the condition...

We develop a spectral-theoretic harmonic analysis for an arbitrary UMD space X. Our approach utilizes the spectral decomposability of X and the multiplier theory for ${L}_{X}^{p}$ to provide on the space X itself analogues of the classical themes embodied in the Littlewood-Paley Theorem, the Strong Marcinkiewicz Multiplier Theorem, and the M. Riesz Property. In particular, it is shown by spectral integration that classical Marcinkiewicz multipliers have associated transforms acting on X.

Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ${\u2099\left(U\right)}_{n=1}^{\infty}$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $\u2099\left(U\right)=1/n{\sum}_{0<\left|k\right|\le n}(1-|k|/(n+1)){U}^{k}$. We show that this sequence ${\u2099\left(U\right)}_{n=1}^{\infty}$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and...

For 1 ≤ q < ∞, let ${}_{q}\left(\right)$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes ${}_{q}\left(\right)$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction q >...

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