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2005 | 167 | 3 | 245-257
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An $M_{q}(𝕋)$-functional calculus for power-bounded operators on certain UMD spaces

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For 1 ≤ q < ∞, let $𝔐_{q}(𝕋)$ 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 $𝔐_{q}(𝕋)$ 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 $𝔐_{q}(𝕋)$-multiplier result shows that if X ∈ ℐ, and U is an invertible power-bounded operator on X, then U has an $𝔐_{q}(𝕋)$-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 $𝓒_{p}$ (1 < p < ∞), as well as all closed subspaces of any UMD lattice of functions on a σ-finite measure space. The $𝔐_{q}(𝕋)$-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.
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  • Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801, U.S.A.
  • School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3JZ, Scotland, U.K.
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