Comparison of spectra of absolutely regular distributions and applications

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 $F\ast \psi \in C_0(\mathring{,}X)$ for all $ψ\in \f(\r)$. 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

arXiv comment24 pages

References in corpus (1)

• Spectral criteria for solutions of evolution equations and comments on reduced spectra