How to declare a delimiter to be a left or right delimiter?

Question

TeX knows the difference between a "left" and a "right" delimiter (e.g. ( and )) and adjusts spacing accordingly. However, the left and right versions of the delimiter | are the same, so TeX has to guess. Usually TeX guesses correctly, but sometimes it does not. Now, I know that I can use \left and \right to force TeX to consider a given | as a left or right delimiter, however these have the additional effect of changing the size (which maybe I don't want). Even the smallest "big" commands \bigl and \bigr increase the size (albeit only a small amount). Is there a way to make | a left or right delimiter without changing its size at all?

Here is a specific example where I know how to do it, but it isn't a general solution:

\documentclass{article}

\usepackage{amsmath,amssymb}

\begin{document}

$|\operatorname{A}|$

$|{\operatorname{A}}|$

$\left|\operatorname{A}\right|$

\end{document}

The spacing is incorrect on the first line and correct on the second and third lines. The second line works well, but seems a bit hacked to me. The third line isn't good since it would increase the size of the delimiters if the stuff in between were taller (and maybe I don't want that).

(Yes, I know that the reason the spacing is incorrect on the first line is because TeX is treating the second | as the "argument" of \operatorname{A}.)

Answer 1

If you say, in a math formula, \mathopen|, then the vertical bar will be interpreted as an Open atom (i.e., a left delimiter), yet its size won’t be changed. Similarly, \mathclose| yields a Close atom (i.e., a right delimiter) without any change in size. See also the answer to Exercise 18.14 in The TeXbook. MWE:

% My standard header for TeX.SX answers:
\documentclass[a4paper]{article} % To avoid confusion, let us explicitly 
                                 % declare the paper format.

\usepackage[T1]{fontenc}         % Not always necessary, but recommended.
% End of standard header.  What follows pertains to the problem at hand.



\begin{document}

For example, \( \mathopen|-x\mathclose| \neq |-x| \).

\end{document}

Notwithstanding this, I think that the solution that @Bernard recommends in a comment, that is, to use \DeclarePairedDelimiter, is The Right Thing To Do.

Edit: As Barbara Beeton reminds in her comment, if the problem is confined to single and double vertical bars (and if you prefer not to load the mathtools package, but stick to amsmath), you can also use the four predefined commands \lvert, \rvert, \lVert, and \rVert: l stands for “left”, r for “right”, lowercase v for single bar, uppercase V for double bar.

For a deeper discussion, see, for example, Using \big| and \right| versus \bigr\rvert and \right\rvert (I remember this question because one of the answers was mine!).