We know that normal operators are "nice". In the finite dimensional case, the spectral theorem tells us everything we need to know. In the infinite dimensional case, we can define a continuous functional calculus over the (bounded) normal operators (admittedly I don't know much about the unbounded case). However, is there a context in which we should "expect" an operator to be normal? That is:
Is there a common application of linear algebra (or operator theory) in which we would expect the operator we're working with to be an arbitrary normal operator?
There are many contexts in which we might expect the operator in question to be self-adjoint (the Hessian matrix for instance), and there are certainly some in which we might expect the operator to be unitary/orthogonal (the orthogonal Procrustes problem comes to mind). However, I'm looking for a "natural" application in which we might expect a normal operator of any sort, but not expect a non-normal operator.
I found this MO thread about applications of the spectral theorem, but that hasn't led to any satisfying leads, unfortunately.
Thanks for any feedback.