Spectral characterization of absolutely regular vector-valued distributions
arXiv:1006.3358
Abstract
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 is a Banach space. We show that if is bounded or slowly oscillating on $\jj$ with $0\not\in sp_{\A,\f} (F)$, where $\A$ is or $C_0 (\jj,X)$ for example and $\f=\f(\r)$, then is ergodic. This result is new even for $F\in BUC(\jj,X)$ and $\A= C_0(\jj,X)$. If 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 for all $ψ\in \f(\r)$. 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.
20 pages