I currently have the following:
\DeclareMathOperator{\A}{A} $1$-$\A$ $1\text{-}\A$ $B_{1\text{-}\A}$ $1$-$\A$ is okay, while $1\text{-}\A$ and $B_{1\text{-}\A}$ puts unnecessary space between "-" and "A". I want to remove this space, but for $B_{1\text{-}\A}$ I cannot simply end and restart the $$ as in $1$-$\A$. How can I do this?
That's expected: with 1\text{-}\A you have
Ord Ord Op
and the rules of TeX insert a thin space between an Ord atom and an Op atom.
My impression is that either
\A to be a math operator, or\DeclareMathOperator{\oneA}{1-A}I've discussed elsewhere the difference between \operatorname (which is what \DeclareMathOperator defines abbreviations of) and \mathrm.
If you have a “family” of operators of the form “*-A”, where * can be anything, you can define
\newcommand{\xA}[1]{\operatorname{#1-A}} so you can call \xA{1}.
If you want to keep the complicated 1\text{-}A (but I recommend you not to), nullify the operator nature of \A:
1\text{-}{\A} 
-and the\text{-}supposed to mean here?-in your document? This is hyphen character which isn't normally used in math formulas. If you mean "minus", use hyphen character without\textand it changes to minus character automatically. What does mean\Ain your document? You declared it as Op (operator), so you want to use it typically with special limits and with spaces around it. You can't be surprise that there is space around it. If you mean it as ordinary character, declare Ord, not Op.