Self-adjointness and boundedness in quadratic quantization

We construct a counter example showing, for the quadratic quantization, the identity $(\mathrm{\Gamma }(T){)}^{\ast }=\mathrm{\Gamma }(T^{\ast })$ is not necessarily true. We characterize all operators on the one-particle algebra whose quadratic quantization are self-adjoint operators on the quadratic Fock space. Finally, we discuss the boundedness of the quadratic quantization.