Year: 2010
Type: preprint
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 $X$ 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 $\frak{F}*ψ\in C_0(\r,X)$ 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.
Source: arXiv (Cornell University)
Authors: Basit, Bolis, Pryde, Alan J.
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W35302616