Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Perturbation of closed range operators and Moore-Penrose inverse   Let $H_1,H_2$ be complex Hilbert spaces and $T:H_1\rightarrow H_2$ be a
densely defined closed operator with domain $D(T)\subseteq H_1$ and
$T^{\dagger}$ be the Moore-Penrose inverse of $T$. Let $S:H_1\rightarrow H_2$
be a bounded operator. In this article we focus our attention on the following
questions:
  $1.$ Under what conditions closedness of range of $T$ will imply the
closedness of range of $T+S$?
  $2.$ What is the relation between $T^{\dagger}$ and $(T+S)^{\dagger}$?
  $3.$ What is the relation between $T^{\dagger}$ and $S^{\dagger}$?.

21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph