# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

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

### Tom 167 / 2005

Studia Mathematica 167 (2005), 245-257 MSC: Primary 42A45, 46B70, 46E40, 47B40. DOI: 10.4064/sm167-3-6

#### Streszczenie

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.

#### Autorzy

• Earl BerksonDepartment of Mathematics
University of Illinois
1409 W. Green St.
Urbana, IL 61801, U.S.A.
e-mail
• T. A. GillespieSchool of Mathematics
University of Edinburgh
James Clerk Maxwell Building
Edinburgh EH9 3JZ, Scotland, U.K.
e-mail

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