What is the difference between the several definitions of mathematic operators?

Question

I am defining a new math operator, and I found that I can define it using 4 different commands:

• \DeclareMathOperator (package amsmath)
• \operatorname (package amsmath)
• \DeclareMathOperator* (package amsmath)
• \mathop (TeX command)

What is the formatting difference between them?

Answer 1

With the MWE below, we see that the main differences are following:

• The operator's font changes: all commands write an upright operator, but \mathop write it slanted.
• In display mode both \DeclareMathOperator* and \mathop set sub- and superscripts right below and above the operator, while the other operators set them shifted to the right.
• The operator stands right on the baseline for all commands but for\mathop there it sinks below it (both in inline and display mode)

See picture and code below:

\documentclass{article}
\usepackage{amsmath}

%All operators are declared as commands
\DeclareMathOperator{\Ti}{T}
\newcommand{\Tii}{\operatorname{T}}
\DeclareMathOperator*{\Tiii}{T}
\newcommand{\Tiv}{\mathop{T}}

\begin{document}

%DeclareMathOperator
\verb|\DeclareMathOperator|\\
Inline:
\(x \Ti^a_b y\)
\\
Display:
\[x \Ti^a_b y\]

%Operatorname
\verb|\operatorname|\\
Inline:
\(x \Tii^a_b y\)
\\
Display:
\[x \Tii^a_b y\]

%DeclareMathOperator*
\verb|\DeclareMathOperator*|\\
Inline:
\(x \Tiii^a_b y\)
\\
Display:
\[x \Tiii^a_b y\]

%Mathop
\verb|\mathop|\\
Inline:
\(x \Tiv^a_b y\)
\\
Display:
\[x \Tiv^a_b y\]

\end{document}