  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 $L^p(X)$.
