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

Earl Berkson. "Spectral theory and operator ergodic theory on super-reflexive Banach spaces." Studia Mathematica 200.3 (2010): 221-246. <http://eudml.org/doc/286635>.

@article{EarlBerkson2010,
abstract = {On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that $sup_\{n∈ ℕ, z∈ \} || ∑_\{0<|k|≤n\} (1 - |k|/(n+1))k^\{-1\}z^\{k\}U^\{k\}|| < ∞$. (*) 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 $V_\{p\}()$ of functions having bounded p-variation, it transpires that every trigonometrically well-bounded operator on a super-reflexive space X has a norm-continuous $V_\{p\}()$-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 $V_\{p\}()$-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.},
author = {Earl Berkson},
journal = {Studia Mathematica},
keywords = {ergodic Hilbert transform; super-reflexive Banach space; spectral decomposition; -variation; trigonometrically well-bounded operator; James inequalities; Young-Stieltjes integration; Tauberian type theorem},
language = {eng},
number = {3},
pages = {221-246},
title = {Spectral theory and operator ergodic theory on super-reflexive Banach spaces},
url = {http://eudml.org/doc/286635},
volume = {200},
year = {2010},
}

TY - JOUR
AU - Earl Berkson
TI - Spectral theory and operator ergodic theory on super-reflexive Banach spaces
JO - Studia Mathematica
PY - 2010
VL - 200
IS - 3
SP - 221
EP - 246
AB - On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that $sup_{n∈ ℕ, z∈ } || ∑_{0<|k|≤n} (1 - |k|/(n+1))k^{-1}z^{k}U^{k}|| < ∞$. (*) 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 $V_{p}()$ of functions having bounded p-variation, it transpires that every trigonometrically well-bounded operator on a super-reflexive space X has a norm-continuous $V_{p}()$-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 $V_{p}()$-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.
LA - eng
KW - ergodic Hilbert transform; super-reflexive Banach space; spectral decomposition; -variation; trigonometrically well-bounded operator; James inequalities; Young-Stieltjes integration; Tauberian type theorem
UR - http://eudml.org/doc/286635
ER -