An $M_q({\Bbb T})$-functional calculus for power-bounded operators on certain UMD spaces

Earl Berkson¹ ; T. A. Gillespie²

¹ Department of Mathematics University of Illinois 1409 W. Green St. Urbana, IL 61801, U.S.A.
² School of Mathematics University of Edinburgh James Clerk Maxwell Building Edinburgh EH9 3JZ, Scotland, U.K.

Studia Mathematica, Tome 167 (2005) no. 3, pp. 245-257

Résumé

For $1\leq q \infty $, let ${{\mathfrak M}}_{q}( {\mathbb 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 ${\mathcal I}$ of UMD spaces such that whenever $X\in {\mathcal I}$, the sequence space $\ell ^{2}( {\mathbb Z},X) $ admits the classes ${{\mathfrak M}}_{q}( {\mathbb 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 ${{\mathfrak M}}_{q}( {\mathbb T}) $-multiplier result shows that if $X\in {\mathcal I}$, and $U$ is an invertible power-bounded operator on $X$, then $U$ has an ${{\mathfrak M}}_{q}( {\mathbb T}) $-functional calculus for an appropriate range of values of $q>1$. The class ${\mathcal I}$ includes, in particular, all closed subspaces of the von Neumann–Schatten $p$-classes ${\mathcal C}_{p}$ ($1 p \infty $), as well as all closed subspaces of any UMD lattice of functions on a $\sigma $-finite measure space. The ${{\mathfrak M}}_{q}( {\mathbb T}) $-functional calculus result for ${\mathcal I}$, when specialized to the setting of closed subspaces of $L^{p}( \mu ) $ ($\mu $ an arbitrary measure, $1 p \infty $), recovers a previous result of the authors.

Détail

DOI : 10.4064/sm167-3-6

Keywords: leq infty mathfrak mathbb denote banach algebra consisting bounded complex valued functions unit circle having uniformly bounded q variation dyadic arcs describe broad class mathcal umd spaces whenever mathcal sequence space ell mathbb admits classes mathfrak mathbb fourier multipliers appropriate range values range depending multiplier result expands vector valued marcinkiewicz multiplier theorem direction moreover taken conjunction vector valued transference mathfrak mathbb multiplier result shows mathcal invertible power bounded operator has mathfrak mathbb functional calculus appropriate range values class mathcal includes particular closed subspaces von neumann schatten p classes mathcal infty closed subspaces umd lattice functions sigma finite measure space mathfrak mathbb functional calculus result mathcal specialized setting closed subspaces arbitrary measure infty recovers previous result authors