Spectral characterization of absolutely regular vector-valued distributions

Details Spectral characterization of absolutely regular vector-valued distributions Spectral characterization of absolutely regular vector-valued distributions

2010-06-17

arxiv.org/abs/1006.3358v1

Mathematics

Functional Analysis

20 pages

Scientific paper

We study the reduced Beurling spectra $sp_{\Cal {A},V} (F)$ of functions $F \in L^1_{loc} (\jj,X)$ relative to certain function spaces $\Cal{A}\st L^{\infty}(\jj,X)$ and $V\st L^1 (\r)$, 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 $0\not\in sp_{\A,\f} (F)$, where $\A$ is $\{0\}$ or $C_0 (\jj,X)$ for example and $\f=\f(\r)$, then $F$ is ergodic. This result is new even for $F\in BUC(\jj,X)$ and $\A= C_0(\jj,X)$. If $F$ is ergodic and belongs to the space $ \f'_{ar}(\jj,X)$ of absolutely regular distributions and if $sp_{C_0(\jj,X),\f} (F)=\emptyset$, then $\frak{F}*\psi \in C_0(\r,X)$ for all $\psi\in \f(\r)$. Here, $\frak{F}|\jj =F$ and $\frak{F}|(\r\setminus\jj) =0$. 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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