Fourier Multipliers in Homogeneous Banach Spaces
Ben de Pagter (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Werner J. Ricker (Katholische Universität Eichstätt - Ingolstadt)
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Abstract
Let G be an infinite, compact abelian group, E be a homogeneous Banach space over G and LE be the space of all continuous linear operators from E into itself equipped with the operator norm. Translation operators are isometries in E (by definition) and so the closed subalgebra mE of LE consisting of those operators which commute with all translations is well defined. It is shown that there exists a contractive projection Q of LE onto mE which is positivity preserving. Moreover, every operator QT∈mE, with T∈LE, is induced by a unique Fourier multiplier function T^∈ℓ∞Γ, where Γ is the dual group of G. In the setting of the homogeneous Banach spaces LpG, for 1≤p<∞ and G an amenable group, these results are due to W. Arendt and J. Voigt.
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