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

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For 1 鈮 q < 鈭, let $饾攼_{q}(饾晪)$ 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 鈩 of UMD spaces such that whenever X 鈭 鈩, the sequence space 鈩撀(鈩,X) admits the classes $饾攼_{q}(饾晪)$ 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 $饾攼_{q}(饾晪)$-multiplier result shows that if X 鈭 鈩, and U is an invertible power-bounded operator on X, then U has an $饾攼_{q}(饾晪)$-functional calculus for an appropriate range of values of q > 1. The class 鈩 includes, in particular, all closed subspaces of the von Neumann-Schatten p-classes $饾搾_{p}$ (1 < p < 鈭), as well as all closed subspaces of any UMD lattice of functions on a 蟽-finite measure space. The $饾攼_{q}(饾晪)$-functional calculus result for 鈩, when specialized to the setting of closed subspaces of $L^{p}(渭)$ (渭 an arbitrary measure, 1 < p < 鈭), recovers a previous result of the authors.
S艂owa kluczowe
  • 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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