Question is pretty simple, here is the first part.
Consider a spin-1/2 particle. I have two polariser kind of device which detects their spin along the axis of polariser. (I am using polariser in a general sense. You can assume two Stern Gerlach Apparatus which somehow let's you measure the spin state successively) Now, if we put our two polarisers one after other, and would like to study what would it give after these two operations. This successively measuring or operation is denoted by a multiplication of operators in linear algebra. Like if we measure A first then B, total operation on your state is BA. Good.
Now the second part.
All physical operations in Quantum Mechanics are Hermitian Operators as given in the postulates. The converse is also true? All Hermitian Operators can be associated to a physical operation but that physical operation may not be very insightful.
Therefore, these A and B, which are polarising actions (maybe pauli spin X and Y matrices) are physically realisable and are Hermitian indeed.
But when we apply them successively, that is BA, is not a Hermitian matrix and hence not physically realisable. But in the first paragraph, we discussed the exact situation where we applied BA.
So, these questions come out from this
What would we see after applying X and Y on a particle in an experiment? We have read series of Stern Gerlach operations many times in books, so it should be possible, right?
Why do we impose hermiticity on observables still, when there are many cases of non-conservative systems having Non-Hermitian operators. Because BA is realisable and also non-commuting (this is an imposition from my side, otherwise the question wouldn't have existed).