It's really necessary for an observable represented by an operator acting in a Hilbert space to be hermitian? It's known that not only hermitian operators have real eigenvalues and that also normal operators can have real eigenvalues, even if not always.
Reading the work of Bender https://arxiv.org/abs/hep-th/0703096 i saw that there's a particular class of operators that have real eigenvalues (Parity-Time invariant operators in the regime of unbroken PT simmetry). Why isn't common to study them? Which is the largest class of operators with real eigenvalues?
It's just a legacy or what?