Say we have a set of (invertible) linear operators $ \{ A, B, C \}$. Is there any meaningful sense in which we could conceive of an ``average'' of these linear operators?
It would seem like the easy first answer would be some version of geometric mean, but for the fact that operators do not commute.
Another idea I thought about would be some (invertible) $D$ s.t. for $A' := A D^{-1}$, $B' := B D^{-1}$, and $C' := C D^{-1}$ we have $A' B' C' = I$, but again I can't see how this works given that operators don't commute.
I believe this question was asked previously here:
Do matrices have average and fluctuations?
But I don't think the response really answered the OP's question.