An $M_{q}(𝕋)$-functional calculus for power-bounded operators on certain UMD spaces

• Autorzy: Earl Berkson , T. A. Gillespie
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46E40: Spaces of vector- and operator-valued functions
• 42A45: Multipliers
• 46B70: Interpolation between normed linear spaces
• 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2005
• Tom: 167
• Numer: 3
• Strony: 245-257

Daty

wydano

2005

Twórcy

autor

Earl Berkson

• Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801, U.S.A.

autor

T. A. Gillespie

• School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3JZ, Scotland, U.K.

Identyfikatory

DOI

10.4064/sm167-3-6