Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Katznelson-Tzafriri theorem for analytic Besov functions of operators   Let $T$ be a power-bounded operator on a Banach space $X$, $\mathcal{A}$ be a
Banach algebra of bounded holomorphic functions on the unit disc $\mathbb{D}$,
and assume that there is a bounded functional calculus for the operator $T$, so
there is a bounded algebra homomorphism mapping functions $f \in \mathcal{A}$
to bounded operators $f(T)$ on $X$. Theorems of Katznelson-Tzafriri type
establish that $\lim_{n\to\infty} \|T^n f(T)\| = 0$ for functions $f \in
\mathcal{A}$ whose boundary functions vanish on the unitary spectrum
$\sigma(T)\cap \mathbb{T}$ of $T$, or sometimes satisfy a stronger assumption
of spectral synthesis. We consider the case when $\mathcal{A}$ is a standard
Banach algebra $\mathcal{B}(\mathbb{D})$ of analytic Besov functions on
$\mathbb{D}$. We set out the relevant properties of the algebra
$\mathcal{B}(\mathbb{D})$, and we define the $\mathcal{B}(\mathbb{D})$-calculus
in an explicit way. We show in particular that the existence of a bounded
$\mathcal{B}(\mathbb{D})$-calculus for $T$ is equivalent to an integral
condition for the resolvent of $T$. The $\mathcal{B}(\mathbb{D})$-calculus has
been briefly touched on in some previous papers, but our approach is new and it
follows the analogous theory for bounded $C_0$-semigroups developed in two
recent papers. This approach enables us to prove a Katznelson-Tzafriri theorem
for the $\mathcal{B}(\mathbb{D})$-calculus which extends several previous
results.

13
The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th