Comparison of spectra of absolutely regular distributions and applications

TL;DR

This paper compares various spectra of functions in Banach spaces, establishing new results on ergodicity and Tauberian theorems, with broader applicability than previous studies.

Contribution

It introduces generalized spectral comparison methods for functions and distributions, extending previous results and providing new criteria for ergodicity and spectral analysis.

Findings

Functions with certain spectral properties are ergodic.

Convolution with test functions yields functions in C_0.

Tauberian theorems follow from spectral analysis.

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 $V\st L^1 (\r)$ and compare them with other spectra including the weak Laplace spectrum. Here $\jj$ is $\r_+$ or $\r$ and $X$ is a Banach space. If $F$ belongs to the space $ \f'_{ar}(\jj,X)$ of absolutely regular distributions and has uniformly continuous indefinite integral with $0\not\in sp_{\A,\f(\r)} (F)$ (for example if F is slowly oscillating and $\A$ is $\{0\}$ or $C_0 (\jj,X)$), then $F$ is ergodic. If $F\in \f'_{ar}(\r,X)$ and $M_h F (\cdot)= \int_0^h F(\cdot+s)\,ds$ is bounded for all $h > 0$ (for example if $F$ is ergodic) and if $sp_{C_0(\r,X),\f} (F)=\emptyset$, then ${F}*\psi \in C_0(\r,X)$ for all $\psi\in \f(\r)$. We show that tauberian theorems for Laplace transforms follow from results about reduced spectra. Our results are more general than previous ones and we demonstrate this through examples