Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rates of decay in the classical Katznelson-Tzafriri theorem   Given a power-bounded operator $T$, the theorem of Katznelson and Tzafriri
states that $\|T^n(I-T)\|\to0$ as $n\to\infty$ if and only if the spectrum
$\sigma(T)$ of $T$ intersects the unit circle $\mathbb{T}$ in at most the point
1. This paper investigates the rate at which decay takes place when
$\sigma(T)\cap\mathbb{T}=\{1\}$. The results obtained lead in particular to
both upper and lower bounds on this rate of decay in terms of the growth of the
resolvent operator $R(\mathrm{e}^{\mathrm{i}\theta},T)$ as $\theta\to0$. In the
special case of polynomial resolvent growth, these bounds are then shown to be
optimal for general Banach spaces but not in the Hilbert space case.

29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat