Spectral theory and operator ergodic theory on super-reflexive Banach spaces

Earl Berkson¹

¹ Department of Mathematics University of Illinois 1409 W. Green Street Urbana, IL 61801, U.S.A.

Studia Mathematica, Tome 200 (2010) no. 3, pp. 221-246

Résumé

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.

Détail

DOI : 10.4064/sm200-3-2

Keywords: reflexive spaces trigonometrically well bounded operators have operator ergodic theory characterization invertible operators sup mathbb mathbb biggl sum leq biggl frac biggr biggr infty tag* * trigonometrically well bounded operators permeate many settings modern analysis note highlights advances their spectral theory operator ergodic theory made possible recent rekindling interest nbsp james inequalities super reflexive spaces james inequalities combined young stieltjes integration spaces mathbb functions having bounded p variation transpires every trigonometrically well bounded operator super reflexive space has norm continuous mathbb functional calculus range values investigate ways outcome logically simplifies simultaneously expands structure theory fourier analysis operator ergodic theory trigonometrically well bounded operators particular super reflexive space general relexive space theorem tauberian type holds averages * corresponding trigonometrically well bounded operator replaced set rotated ergodic hilbert averages which precompact set relative strong operator topology circle ideas facilitated development convergence theorem nets spectral integrals mathbb functions hilbert space setting apply foregoing operator weighted shifts which known provide universal model trigonometrically well bounded operators hilbert space