 

Earl Berkson
Rotation methods in operator ergodic theory view print


Published: 
February 7, 2011 
Keywords: 
spectral decomposition, trigonometrically wellbounded operator, higher variation, Fourier series 
Subject: 
Primary 26A45, 42A16, 47B40 


Abstract
Let E(⋅):R→B(X)
be the spectral decomposition of a trigonometrically
wellbounded operator U acting on the arbitrary Banach space X,
and suppose that the bounded function ϕ:T→ C
has the property that for each z∈T, the spectral integral
∫_{[0,2π]}ϕ(e^{it}) dE_{z}(t)
exists, where E_{z}(⋅) denotes the spectral
decomposition of the (necessarily) trigonometrically wellbounded operator
(zU). We show this implies that for each z∈ T,
the spectral integral with respect to E(⋅) of the
rotated function ϕ_{z}(⋅) ≡ ϕ((⋅)z) exists. In particular, these considerations furnish
the preservation under rotation of spectral integration for the
Marcinkiewicz rclasses of multipliers M_{r}(T),
which are not themselves rotationinvariant. In the setting of an
arbitrary superreflexive space, we pursue a different aspect of the impact
of rotations on the operator ergodic theory framework by applying the
rotation group to the spectral integration of functions of higher variation
so as to obtain strongly convergent Fourier series expansions for the
operator theory counterparts of such functions. This vectorvalued Fourier
series convergence can be viewed as an extension of classical
CalderónCoifmanWeiss transference without being tied to the need of the latter
for powerboundedness assumptions.


Author information
Department of Mathematics; University of Illinois; 1409 W. Green Street; Urbana, IL 61801 USA
berkson@math.uiuc.edu

