Let $L$ be an operator that maps any well-behaved (with bounded integral) function $f:\mathbb{R} \rightarrow\mathbb{R}$ to a number $L_f\in\mathbb{R}$, such that if $g(x)=f(x-d)$ for all $x\in\mathbb{R}$, then $L_g=L_f+d$.
Subject to the above condition, $L_f$ could be, for example:
- the centroid $\frac{\int_{-\infty}^\infty xf(x)dx}{\int_{-\infty}^\infty f(x)dx}$
- $\inf \arg \max_x{f(x)}$
- the median
- $\inf \arg \max_x{f'(x)}$
or any other domain point serving as some kind of "anchor" to the function's "signature", such that it "moves along" when the function is translated.
Is there a name for such a property/operator in general?
I don't suspect there is a standard term, but I might call such a map $L$ horizontal-translation-preserving because it preserves horizontal translation of the input function by mapping it to translation of the output real number. Formally, $$ L: (\mathbb{R} \to \mathbb{R}) \to \mathbb{R} $$ preserves horizontal translation (also called "shifting") if it satisfies $$ L(x \mapsto f(x - d)) = L(f) + d. $$
(I would prefer just "translation-preserving", but one has to distinguish between translation in the $x$- and $y$-direction of a function.)
Note that such a map $L$ need not be a linear operator (see e.g. the $\inf \arg \max$ example), and it also need not be a bounded linear operator, which are the objects normally studied in functional analysis. So I would be careful when saying "let $L$ be an operator" that you clarify you don't mean linear or bounded linear.