Operator space trace class operators

First we define the linear map that produces the densities from N-particle states. It is a map from the space of A -particle Trace Class operators into the space of complex valued absolute integrable functions of space-spin variables... 

The space of trace class operators acting in the Hilbert space... 

K)/ /KerE N Linear space of equivalence classes of Trace Class operators. The operators are equivalent if there difference lies in the kernel of... 

Eh Linear energy functional based on the Hamiltonian H it acts on the space of A-particle Trace Class operators. 

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. 

Density matrices can be characterized as trace-class linear operators on a Hilbert space, > —> >, such that is self-adjoint, non-negative and of trace equal to one, where trace is defined by... 

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... 

Exercise 2.30 Define an equivalence of matrices by Aj A2 if and only if there is a matrix B such that Ai = BA 2,B. Show that matrix multiplication is well defined on equivalence classes. Shoyv that trace and determinant are well defined on equivalence classes. Show that eigenvalues are well defined, but eigenvectors are not. Finally, show that given a vector space V, any linear operator on V corresponds to precisely one equivalence class of matrices. Exercise 2.31 Suppose V is a finite-dimensional vector space. 

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