# New Estimates on the bounds of Brunel's operator.

@article{Assani2020NewEO, title={New Estimates on the bounds of Brunel's operator.}, author={Idris Assani and R. Spencer Hallyburton and Sebastien McMahon and Stefano Schmidt and Cornelis Jan Schoone}, journal={arXiv: Dynamical Systems}, year={2020} }

We study the coefficients of the Taylor series expansion of powers of the function $\psi(x)=\frac{1-\sqrt{1-x}}{x}$, where Brunel's operator $A$ is defined as $\psi(T)$. The operator $A$ was shown to map positive mean-bounded (and power-bounded) operators to positive power-bounded operators. We provide specific details of results announced by A. Brunel and R. Emilion in \cite{Brunel}. In particular, we sharpen an estimate to prove that $\sup_{n\in\mathbb{N}} \|n(A^n-A^{n+1})\| < \infty$. We…

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