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

• Autorzy: Earl Berkson
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 26A45: Functions of bounded variation, generalizations
• 47A35: Ergodic theory
• 46B20: Geometry and structure of normed linear spaces
• 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 200
• Numer: 3
• Strony: 221-246

Daty

wydano

2010

Twórcy

autor

Earl Berkson

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

Identyfikatory

DOI

10.4064/sm200-3-2