Spectral characterization of absolutely regular vector-valued distributions

TL;DR

This paper investigates the spectral properties of vector-valued distributions and their implications for ergodicity and Tauberian theorems, extending previous results with broader applicability and new insights.

Contribution

It introduces new spectral criteria for ergodicity of vector-valued distributions and connects these to Tauberian theorems, broadening the scope of prior work.

Findings

If $F$ is bounded or slowly oscillating with $0 otin sp_{\\\ ext{A},\f}(F)$, then $F$ is ergodic.

Ergodic $F$ in certain spaces implies convolution results in $C_0( ,X)$.

The spectral approach yields more general Tauberian theorems for Laplace transforms.

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 $\frak{F}*\psi \in C_0(\r,X)$ 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.