Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The essential spectrum, norm, and spectral radius of abstract
  multiplication operators   Let $E$ be a complex Banach lattice and $T$ is an operator in the centrum
$Z(E)=\{T: |T|\le \lambda I \mbox{ for some } \lambda\}$ of $E$. Then the
essential norm $\|T\|_{e}$ of $T$ equals the essential spectral radius
$r_{e}(T)$ of $T$. We also prove $r_{e}(T)=\max\{\|T_{A^{d}}\|,
r_{e}(T_{A})\}$, where $T_{A}$ is the atomic part of $T$ and $T_{A^{d}}$ is the
non-atomic part of $T$. Moreover $r_{e}(T_{A})=\limsup_{\mathcal
F}\lambda_{a}$, where $\mathcal F$ is the Fr\'echet filter on the set $A$ of
all positive atoms in $E$ of norm one and $\lambda_{a}$ is given by
$T_{A}a=\lambda_{a}a$ for all $a\in A$.

7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int