Self-adjointness and boundedness in quadratic quantization

We construct a counter example showing, for the quadratic quantization, the identity $(Γ(T))^*= Γ(T^*)$ 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.

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