Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on property (gb) and perturbations   An operator $T \in \mathcal{B}(X)$ defined on a Banach space $X$ satisfies
property $(gb)$ if the complement in the approximate point spectrum
$\sigma_{a}(T)$ of the upper semi-B-Weyl spectrum $\sigma_{SBF_{+}^{-}}(T)$
coincides with the set $\Pi(T)$ of all poles of the resolvent of $T$. In this
note we continue to study property $(gb)$ and the stability of it, for a
bounded linear operator $T$ acting on a Banach space, under perturbations by
nilpotent operators, by finite rank operators, by quasi-nilpotent operators
commuting with $T$. Two counterexamples show that property $(gb)$ in general is
not preserved under commuting quasi-nilpotent perturbations or commuting finite
rank perturbations.

37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph