抽象发展方程无界解的多项式遍历性和渐近性
Polynomial ergodicity and asymptotic behaviour of unbounded solutions of abstract evolution equations
摘要：
在本文中，我们发展的遍历性的概念，包括功能占主导地位的重量$w$。这些函数具有多项式平均值，并且包括$w$ -几乎周期函数。这使我们能够描述线性发展、递归和卷积方程的无界解的渐近行为。为了统一的治疗，并允许进一步的应用，我们考虑解决方案$ϕ: G\rightarrow X$的广义发展方程的形式$ (*) (Bϕ)(t)=Aϕ(t)+ψ(t)$为$t\in G$其中$G$ \是一个局部紧阿贝尔群与一个封闭的子半群$J$，$A$是一个封闭的线性算子的Banach空间$X$，$ψ:G\rightarrow X$是连续的，$B$是一个线性算子的特征函数$θ_{B}:\hat{G}\rightarrow \mathbf{C}$。我们引入共振集$θ_{B}^{-1}(σ(A))$，它包含齐次方程$Bϕ=A\circ ϕ$所有解的Beurling谱.对于某些类${\F} $的功能从$J$ $% X,$的谱$sp_{\F}(ϕ)$的$ϕ$相对于$% {\F} $是用来确定成员的${\F}.$我们的主要结果给出了一般条件下，$sp_{\F}(ϕ)$ \是一个子集的共振集。作为一个简单的结果，我们得到的条件下，$ψ|_{J}\in \F$意味着$ϕ|_{J}\in {\F}.$一个重要的工具是我们推广到无界函数的卢米斯定理。作为应用，我们得到了许多已知结果的推广或新的证明，包括Gelfand，Hille，Katznelson-Tzafriri，Esterle等人的定理，冯，鲁斯和阿伦特-巴蒂。
英文摘要：
In this paper we develop the notion of ergodicity to include functions dominated by a weight $w$. Such functions have polynomial means and include, amongst many others, the $w$-almost periodic functions. This enables us to describe the asymptotic behaviour of unbounded solutions of linear evolution, recurrence and convolution equations. To unify the treatment and allow for further applications, we consider solutions $ϕ: G\rightarrow X$ of generalized evolution equations of the form $ (*) (Bϕ)(t)=Aϕ(t)+ψ(t)$ for $t\in G$ where $G$\ is a locally compact abelian group with a closed subsemigroup $J$, $A$ is a closed linear operator on a Banach space $X$, $ψ:G\rightarrow X$ is continuous and $B$ is a linear operator with characteristic function $θ_{B}:\hat{G}\rightarrow \mathbf{C}$. We introduce the resonance set $θ_{B}^{-1}(σ(A))$ which contains the Beurling spectra of all solutions of the homogeneous equation $Bϕ=A\circ ϕ$. For certain classes ${\F} $ of functions from $J$ to $% X,$ the spectrum $sp_{\F}(ϕ)$ of $ϕ$ relative to $% {\F} $ is used to determine membership of ${\F}.$ Our main result gives general conditions under which $sp_{\F}(ϕ)$\ is a subset of the resonance set. As a simple consequence we obtain conditions under which $ψ|_{J}\in \F$ implies $ϕ|_{J}\in {\F}.$ An important tool is our generalization to unbounded functions of a theorem of Loomis. As applications we obtain generalizations or new proofs of many known results, including theorems of Gelfand, Hille, Katznelson-Tzafriri, Esterle et al., Phóng, Ruess and Arendt-Batty.
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