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\documentclass[12pt, reqno]{amsart}
\usepackage{ amsmath,amsthm, amscd, amsfonts, amssymb, graphicx, color}
\usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref}
\usepackage[running]{lineno}
\textwidth 12 cm \textheight 18 cm

\oddsidemargin 2.12cm \evensidemargin 1.8cm


\setcounter{page}{1}

%------------------------------------------------------------------------------------%

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{\bf{Remark}}
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%------------------------------------------------------------------------------------%



\begin{document}
\linenumbers
\title{Jacobson's Lemma for the generalized n-strongly Drazin inverse}

\author{Huanyin Chen}
\author{Marjan Sheibani}


\begin{abstract}
Let $n\in {\Bbb N}$. An element $a\in R$ has generalized n-strongly Drazin inverse if there exists $x\in R$ such that $xax=x, x\in comm^2(a), a^n-ax\in R^{qnil}.$
For any $a,b\in R$, we prove that $1-ab$ has generalized n-strongly Drazin inverse if and only if $1-ba$ has generalized n-strongly Drazin inverse. Extensions in Banach algebra are also obtained.
\end{abstract}

\maketitle

\section{Introduction}

Let $R$ be an associative ring with an identity. The commutant of $a\in R$ is defined by $comm(a)=\{x\in
R~|~xa=ax\}$. The double commutant of $a\in R$ is defined
by $comm^2(a)=\{x\in R~|~xy=yx~\mbox{for all}~y\in comm(a)\}$. Set $R^{qnil}=\{a\in R~|~1+ax\in U(R)~\mbox{for
every}~x\in comm(a)\}$. An element $a\in R$ is quasinilpotent if
$a\in \grave{R^{qnil}}$. For a Banach algebra $\mathcal{A}$ it is well known
that $$a\in \mathcal{A}^{qnil}\Leftrightarrow
\lim\limits_{n\to\infty}\parallel a^n\parallel^{\frac{1}{n}}=0.$$
An element $a$ in $R$ is said to have generalized Drazin inverse if there exists $x\in R$ such that $$x=xax, x\in comm^2(a), a-a^2x\in R^{qnil}.$$  The preceding $x$ is unique, if such element exists. As usual,
it will be denoted by $a^{d}$, and called the generalized Drazin inverse of $a$. Generalized Drazin inverse is extensively studied in matrix theory and Banach algebra (see~\cite{B, J, LZ} and~\cite{ZC}).

\end{document}


enter image description here

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  • 2
    unrelated but \bf has not been defined by default in latex since 1993, in document classes where it is defined for compatibility with old documents the syntax is {\bf Proof} not \bf{Proof} Commented Dec 26, 2021 at 14:27

1 Answer 1

2

$$ is not LaTeX, if you use the LaTeX syntax for display math then it works as you expect. Also avoid deprecated markup such as \bf and \Bbb

enter image description here

\documentclass[12pt, reqno]{amsart}
\usepackage{ amsmath,amsthm, amscd, amsfonts, amssymb, graphicx, color}
\usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref}
\usepackage[running]{lineno}
\textwidth 12 cm \textheight 18 cm

\oddsidemargin 2.12cm \evensidemargin 1.8cm


\setcounter{page}{1}

%------------------------------------------------------------------------------------%

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{\bf{Remark}}
\newtheorem{alg}{\bf{Algorithm}}
\newtheorem{pf}[thm]{\bf{Proof}}
\newtheorem{exam}[thm]{Example}
\numberwithin{equation}{section}
\def\pn{\par\noindent}
\def\cen{\centerline}
%------------------------------------------------------------------------------------%



\begin{document}
\linenumbers
\title{Jacobson's Lemma for the generalized n-strongly Drazin inverse}

\author{Huanyin Chen}
\author{Marjan Sheibani}


\begin{abstract}
Let $n\in \mathbb{N}$. An element $a\in R$ has generalized n-strongly Drazin inverse if there exists $x\in R$ such that $xax=x, x\in comm^2(a), a^n-ax\in R^{qnil}.$
For any $a,b\in R$, we prove that $1-ab$ has generalized n-strongly Drazin inverse if and only if $1-ba$ has generalized n-strongly Drazin inverse. Extensions in Banach algebra are also obtained.
\end{abstract}

\maketitle

\section{Introduction}
Let $R$ be an associative ring with an identity. The commutant of $a\in R$ is defined by $comm(a)=\{x\in
R~|~xa=ax\}$. The double commutant of $a\in R$ is defined
by $comm^2(a)=\{x\in R~|~xy=yx~\mbox{for all}~y\in comm(a)\}$. Set $R^{qnil}=\{a\in R~|~1+ax\in U(R)~\mbox{for
every}~x\in comm(a)\}$. An element $a\in R$ is quasinilpotent if
$a\in \grave{R^{qnil}}$. For a Banach algebra $\mathcal{A}$ it is well known
that \[a\in \mathcal{A}^{qnil}\Leftrightarrow
\lim\limits_{n\to\infty}\parallel a^n\parallel^{\frac{1}{n}}=0.\]
An element $a$ in $R$ is said to have generalized Drazin inverse if there exists $x\in R$ such that \[x=xax, x\in comm^2(a), a-a^2x\in R^{qnil}.\] The preceding $x$ is unique, if such element exists. As usual,
it will be denoted by $a^{d}$, and called the generalized Drazin inverse of $a$. Generalized Drazin inverse is extensively studied in matrix theory and Banach algebra (see~\cite{B, J, LZ} and~\cite{ZC}).

\end{document}
2
  • 1
    unrelated but never use math italic for words, comm should be \mathrm{comm} or probably better \comm defined as \DeclareMathOperator\comm{comm} Commented Dec 26, 2021 at 14:35
  • 2
    Additionally if the display math needs to be numbered as well you can use \usepackage[mathlines]{lineno}.
    – Marijn
    Commented Dec 26, 2021 at 14:54

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