Spectral theory and operator ergodic theory on super-reflexive Banach spaces
Volume 200 / 2010
Abstract
On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators $U$ such that $$ \sup_{n\in \mathbb{N},\,z\in \mathbb{T}}\,\biggl\| \sum_{0< | k| \leq n}\biggl( 1-\frac{| k| }{n+1}\biggr) k^{-1}z^{k}U^{k}\biggr\| <\infty . \tag*{$(*)$} $$ 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}( \mathbb{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}( \mathbb{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}( \mathbb{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.
