The motivation for this question is related to the link: Is there any differential equation whose solution is the function itself? The question in the link arises as I was looking for an answer in the https://dsp.stackexchange.com/questions/55931/is-match-filter-a-solution-of-some-differential-equation . The answer obtained in the dsp site leads me to ask the question in this math site.
Now the question can we have a $L$ whose solution is Wiener filter. By Wiener filter I mean: https://en.wikipedia.org/wiki/Wiener_filter