Declaremathoperator with language dependence

In my opinion the definition should be so that the operator doesn't change across a document even through language changes.

So if the main language is German, \ker should resolve to “Kern” also in English. However, it's easy to provide also a “variable” kind.

\documentclass{scrartcl}
\usepackage[english,ngerman]{babel}
\usepackage{amsmath}
\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand\DeclareBabelMathOperator{mmO{???}}
 {
  \DeclareMathOperator{#1}
   {
    \str_case:xnF { \use:c { bbl@main@language } } { #2 } { #3 }
   }
 }
\NewDocumentCommand\DeclareVariableBabelMathOperator{mmO{???}}
 {
  \DeclareMathOperator{#1}
   {
    \str_case:xnF { \languagename }{ #2 } { #3 }
   }
 }
\cs_generate_variant:Nn \str_case:nnF { x }
\ExplSyntaxOff


\DeclareBabelMathOperator{\range}{
  {english}{ran}
  {ngerman}{Bild}
}
\DeclareBabelMathOperator{\kernel}{
  {english}{ker}
  {ngerman}{Kern}
}[ker]
\DeclareVariableBabelMathOperator{\vrange}{
  {english}{ran}
  {ngerman}{Bild}
}
\DeclareVariableBabelMathOperator{\vkernel}{
  {english}{ker}
  {ngerman}{Kern}
}[ker]

\begin{document}

\section{Fixed names}
\[
  \dim(V) = \dim(\range(T)) + \dim(\kernel(T))
\]
\begin{otherlanguage*}{english}
\[
  \dim(V) = \dim(\range(T)) + \dim(\kernel(T))
\]
\end{otherlanguage*}

\section{Variable names}
\[
  \dim(V) = \dim(\vrange(T)) + \dim(\vkernel(T))
\]
\begin{otherlanguage*}{english}
\[
  \dim(V) = \dim(\vrange(T)) + \dim(\vkernel(T))
\]
\end{otherlanguage*}

\end{document}

The optional argument is for the string to be used with a language not included in the previous choices.

Here is the output if english and ngerman are swapped throughout.

\documentclass{scrartcl}
\usepackage[german,english]{babel}
\usepackage{amsmath}

\DeclareMathOperator{\range}{\rangename}
\DeclareMathOperator{\kernel}{\kernelname}

\addto\extrasenglish{%
\def\rangename{range}%
\def\kernelname{ker}%
}

\addto\extrasgerman{%
\def\rangename{Bild}%
\def\kernelname{Kern}%
}
\begin{document}

\selectlanguage{english}
english
\begin{align}
  T &: V \rightarrow W \\
  \dim(V) &= \dim(\range(T)) + \dim(\kernel(T))
\end{align}

\selectlanguage{german}
Und jetzt etwas auf Deutsch
\begin{align}
  T &: V \rightarrow W \\
  \dim(V) &= \dim(\range(T)) + \dim(\kernel(T))
\end{align}
\end{document}

I would stick to good old \operatorname and combine it with babel's \extrasLANGUAGE:

\documentclass{scrartcl}
\usepackage[german,english]{babel}
\usepackage{amsmath}

\addto\extrasenglish{%
    \def\range{\operatorname{ran}}%
    \def\kernel{\operatorname{ker}}%
}

\addto\extrasgerman{%
    \def\range{\operatorname{Bild}}%
    \def\kernel{\operatorname{Kern}}%
}
\begin{document}

English
\begin{align}
  T &: V \rightarrow W \\
  \dim(V) &= \dim(\range(T)) + \dim(\kernel(T))
\end{align}

\selectlanguage{german}
Und jetzt etwas auf Deutsch
\begin{align}
  T &: V \rightarrow W \\
  \dim(V) &= \dim(\range(T)) + \dim(\kernel(T))
\end{align}
\end{document}