An $M_{q}()$-functional calculus for power-bounded operators on certain UMD spaces

Earl Berkson ; T. A. Gillespie

Studia Mathematica, Tome 166 (2005), p. 245-257 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2005-01-01

Zbl 1076.47010

EUDML-ID : urn:eudml:doc:284828