Operator space Hilbert-Schmidt operators

A superoperator G defined on the Hilbert-Schmidt operator space is said to be bounded, if it satisfies the inequality... 

Appendix C). The space of Hilbert-Schmidt operators, 2 > is a Hilbert... 

The mixed state TDDFT of Rajagopal et al. (38) differs from our formulation in the aspects mentioned alx)ve and in the nature of the operator space where the supervectors reside. A particularly notable distinction is in the use of the factorization D = QQ of the state density operator that leads to unconstrained variation over the space of Hilbert-Schmidt operatOTS, rather than to a constrained variaticxi of the space of Trace-Class operators. 

The classical operator space, which has been thoroughly investigated in mathematics almost since the beginning of this century, is the Hilbert-Schmidt (HS) space A consisting of all operators A for which the product A fA is a trace-class operator. A review of the use of the HS space as a carrier space for the superoperators in quantum theory has recently been given.25... 

It should be observed that the GNS space A contains the Hilbert-Schmidt space as a subspace, but also that it is considerably larger, since it is now sufficient to require that the product AT1/2 should be a Hilbert-Schmidt operator. As an example, we note that the identity operator 1 belongs to the GNS space but not to the HS space. By using (3.17) and (3.48), one may now write the commutator binary product (A T) in the form ... 

Let us now study the ket-bra operators )(4> formed from the wave functions = B, B < . One knows that, if one goes over to the corresponding orthonormal basis

operator space (which is orthonormal in terms of the Hilbert-Schmidt binary product) is of order pi x p = pc- One has further... 

---