An $M_q({\Bbb T})$-functional calculus for power-bounded operators on certain UMD spaces
Volume 167 / 2005
Abstract
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.
