Is it possible to construct a well defined inner-product (and therefore orthonormality) within the set of self-adjoint trace-class linear operators? In the affirmative case, dynamics could be analyzed in Hilbert space, which seem way more simple that Banach spaces.
Which is the fundamental reason why this is not possible?
It is possible indeed !! It is called Hilbert Schmidt scalar product, it is defined in a Hilbert space of bounded compact operators including trace class operators.
$$\langle A|B\rangle := tr(A^\dagger B)\:.$$
The space of Hilbert Schmidt operators is made of all bounded operators $A$ in the considered Hilbert space, such that $A^\dagger A$ is trace class. It is in fact possible to reformulate all QM using that notion.
