论文标题
$ l^p $空间中的广义双曲和阴影
Generalized Hyperbolicity and Shadowing in $L^p$ spaces
论文作者
论文摘要
众所周知，双曲线操作员具有阴影属性。在有限的尺寸Banach空间的环境中，具有阴影特性等于双曲线。 2018年，Bernardes等人。构建了一个具有阴影属性的操作员，该属性不是双曲线，并解决了一个空旷的问题。在此过程中，他们介绍了一类运算符，这些操作员被称为广义双曲线操作员。这类操作员似乎是双曲线和阴影属性之间的重要桥梁。在本文中，我们表明，对于$ l^p（x）$的大型自然类运算师，广义双曲线和阴影属性的概念一致。我们通过为某些类别的操作员提供足够和必要的条件来拥有阴影属性来做到这一点。我们还介绍了计算工具，该工具允许有或没有阴影属性的运算符构建。利用这些工具，我们展示了某些自然概率分布（例如拉普拉斯分布和库奇分布）如何导致运营商有或没有$ l^p（x）$的阴影属性。
It is rather well-known that hyperbolic operators have the shadowing property. In the setting of finite dimensional Banach spaces, having the shadowing property is equivalent to being hyperbolic. In 2018, Bernardes et al. constructed an operator with the shadowing property which is not hyperbolic, settling an open question. In the process, they introduced a class of operators which has come to be known as generalized hyperbolic operators. This class of operators seems to be an important bridge between hyperbolicity and the shadowing property. In this article, we show that for a large natural class of operators on $L^p(X)$ the notion of generalized hyperbolicity and the shadowing property coincide. We do this by giving sufficient and necessary conditions for a certain class of operators to have the shadowing property. We also introduce computational tools which allow construction of operators with and without the shadowing property. Utilizing these tools, we show how some natural probability distributions, such as the Laplace distribution and the Cauchy distribution, lead to operators with and without the shadowing property on $L^p(X)$.
