加权平移半群：多变量情形
Weighted Translation Semigroups: Multivariable Case
摘要：
M. Embry和A.兰伯特发起了研究的加权翻译半群$\{S_t\}$在${\cal B}(L^2({\mathbb R_+})),$，以期探讨一个连续的模拟加权移位算子。在此基础上，我们刻画了一些特殊类型的半群，并建立了左可逆加权平移半群的一个分析模型。本文讨论了多元集上的加权平移半群的推广。我们发展了分析模型的环面模拟，并描述了光谱图像。我们给出了多变量情况下的加权平移半群的许多例子。进一步，我们用$L^2({\mathbb R_+^d})$代替空间$L^2({\mathbb R_+})$，研究了${\cal B}(L^2({\mathbb R_+^d})),$中加权平移半群$\{S_{\bar{t}}\}$在单变量和多变量情形下的性质.
英文摘要：
M. Embry and A. Lambert initiated the study of a weighted translation semigroup $\{S_t\}$ in ${\cal B}(L^2({\mathbb R_+})),$ with a view to explore a continuous analogue of a weighted shift operator. We continued the work, characterized some special types of semigroups and developed an analytic model for the left invertible weighted translation semigroup. The present paper deals with the generalization of the weighted translation semigroup in multi-variable set up. We develop the toral analogue of the analytic model and also describe the spectral picture. We provide many examples of weighted translation semigroups in multi-variable case. Further, we replace the space $L^2({\mathbb R_+})$ by $L^2({\mathbb R_+^d})$ and explore the properties of weighted translation semigroup $\{S_{\bar{t}}\}$ in ${\cal B}(L^2({\mathbb R_+^d})),$ in both one and multi variable cases.
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