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 $L^p(X)$.

众所周知，双曲线算子具有阴影特性。在有限维Banach空间的设置中，具有阴影性质就等价于双曲线。2018 年，Bernardes 等人。构造了一个具有非双曲线阴影属性的运算符，解决了一个悬而未决的问题。在这个过程中，他们引入了一类被称为广义双曲算子的算子。这类运算符似乎是双曲性和阴影属性之间的重要桥梁。在这篇文章中，我们展示了对于一个大型的自然类算子${升}^{磷}(X)$广义双曲线的概念和阴影属性是一致的。我们通过为特定类别的运算符提供具有阴影属性的充分必要条件来做到这一点。我们还介绍了计算工具，这些工具允许构造带有和不带有阴影属性的运算符。利用这些工具，我们展示了一些自然概率分布（例如拉普拉斯分布和柯西分布）如何导致具有和不具有阴影属性的算子${升}^{磷}(X)$.