New Estimates on the bounds of Brunel's operator

We study the coefficients of the Taylor series expansion of powers of the function $\psi(x)=\frac{1-\sqrt{1-x}}{x}$, where the Brunel operator $A\equiv A(T)$ is defined as $\psi(T)$ for any mean-bounded $T$. We prove several new precise estimates regarding the Taylor coefficients of $\psi^n$ for $n\in\mathbb{N}$. We apply these estimates to give an elementary proof that for any mean-bounded, not necessarily positive operator $T$ on a Banach space $X$, the Brunel operator $A(T):X\to X$ is power-bounded and satisfies $\sup_{n\in\mathbb{N}} \|n(A^n-A^{n+1})\| < \infty$ (equivalently, $A(T)$ is a Ritt operator). Along the way we provide specific details of results announced by A. Brunel and R. Emilion in \cite{Brunel}.