Spectral characterization of absolutely regular vector-valued distributions

2010/06/17 by Bolis Basit, Basit, Bolis, Alan J. Pryde +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1006.3358

20 pages

arxiv created 2010/06/17 · arxiv updated 2010/06/18

Abstract

We study the reduced Beurling spectra sp_\Cal A,V (F) of functions F ∈ L¹_loc (\jj,X) relative to certain function spaces \CalA\st L^∞(\jj,X) and V\st L¹ (\r), where \jj is \r₊ or \r and X is a Banach space. We show that if F is bounded or slowly oscillating on \jj with 0\not∈ sp_\A,\f (F), where \A is \0\ or C₀ (\jj,X) for example and \f=\f(\r), then F is ergodic. This result is new even for F∈ BUC(\jj,X) and \A= C₀(\jj,X). If F is ergodic and belongs to the space \f'_ar(\jj,X) of absolutely regular distributions and if sp_C0(\jj,X),\f (F)=∅, then \frakF*ψ∈ C₀(\r,X) for all ψ∈ \f(\r). Here, \frakF|\jj =F and \frakF|(\r∖\jj) =0. We show that tauberian theorems for Laplace transforms follow from results about the reduced spectrum. Our results are more widely applicable than those of previous authors. We demonstrate this and the sharpness of our results through examples.

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