Where to find examples of mean-bounded operators?

Question

An operator $T$ is said to be mean-bounded if: $$\sup_{n\in\mathbb{N}}\left\Vert\frac{1}{n}\sum_{p=0}^{n-1}T^p\right\Vert<\infty.$$ This is a less restrictive analogue of the property of power-boundedness, in which $\Vert T^n\Vert$ is bounded above. I am looking for a reference with some examples of non-trivial mean-bounded operators (particularly on $L^p$ spaces); I unfortunately don't yet have the functional analysis background to be able to easily construct such operators, but they are necessary in my research and I would like to have some examples to test on.