An $M_q\left(\right)$-functional calculus for power-bounded operators on certain UMD spaces

Earl Berkson, and T. A. Gillespie. "An $M_{q}()$-functional calculus for power-bounded operators on certain UMD spaces." Studia Mathematica 167.3 (2005): 245-257. <http://eudml.org/doc/284828>.

@article{EarlBerkson2005,
abstract = {For 1 ≤ q < ∞, let $_\{q\}()$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes $_\{q\}()$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction q > 1. Moreover, when taken in conjunction with vector-valued transference, this $_\{q\}()$-multiplier result shows that if X ∈ ℐ, and U is an invertible power-bounded operator on X, then U has an $_\{q\}()$-functional calculus for an appropriate range of values of q > 1. The class ℐ includes, in particular, all closed subspaces of the von Neumann-Schatten p-classes $_\{p\}$ (1 < p < ∞), as well as all closed subspaces of any UMD lattice of functions on a σ-finite measure space. The $_\{q\}()$-functional calculus result for ℐ, when specialized to the setting of closed subspaces of $L^\{p\}(μ)$ (μ an arbitrary measure, 1 < p < ∞), recovers a previous result of the authors.},
author = {Earl Berkson, T. A. Gillespie},
journal = {Studia Mathematica},
keywords = {UMD space; multiplier; complex interpolation; -variation; spectral decomposition; spectral integral},
language = {eng},
number = {3},
pages = {245-257},
title = {An $M_\{q\}()$-functional calculus for power-bounded operators on certain UMD spaces},
url = {http://eudml.org/doc/284828},
volume = {167},
year = {2005},
}

TY - JOUR
AU - Earl Berkson
AU - T. A. Gillespie
TI - An $M_{q}()$-functional calculus for power-bounded operators on certain UMD spaces
JO - Studia Mathematica
PY - 2005
VL - 167
IS - 3
SP - 245
EP - 257
AB - For 1 ≤ q < ∞, let $_{q}()$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes $_{q}()$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction q > 1. Moreover, when taken in conjunction with vector-valued transference, this $_{q}()$-multiplier result shows that if X ∈ ℐ, and U is an invertible power-bounded operator on X, then U has an $_{q}()$-functional calculus for an appropriate range of values of q > 1. The class ℐ includes, in particular, all closed subspaces of the von Neumann-Schatten p-classes $_{p}$ (1 < p < ∞), as well as all closed subspaces of any UMD lattice of functions on a σ-finite measure space. The $_{q}()$-functional calculus result for ℐ, when specialized to the setting of closed subspaces of $L^{p}(μ)$ (μ an arbitrary measure, 1 < p < ∞), recovers a previous result of the authors.
LA - eng
KW - UMD space; multiplier; complex interpolation; -variation; spectral decomposition; spectral integral
UR - http://eudml.org/doc/284828
ER -