Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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}.

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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: nlin
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The categories of this paper are: hep-th
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The categories of this paper are: funct-an